prr.hec.gov.pkprr.hec.gov.pk/.../6666/...maths_2015_hitectaxila.pdfiv hitec university taxila the...

Modified )/( GG -Expansion Methods for Soliton

Solutions of Nonlinear Differential Equations

By

Muhammad Shakeel

10-PhD-MT-03

PhD Thesis

In

Mathematics

HITEC University Taxila

Taxila, Pakistan

2015

ii


Modified )/( GG -Expansion Methods for Soliton Solutions of Nonlinear Differential Equations

A Thesis Presented to


In partial fulfillment

of the requirements for the degree of

PhD Mathematics

By

Muhammad Shakeel

10-PhD-MT-03

2015

iii

Modified )/( GG -Expansion Methods for Soliton Solutions of Nonlinear Differential Equations

A thesis submitted to the Department of Mathematics as partial fulfillment of the requirements for the award of Degree of PhD Mathematics.

Name Registration Number

Muhammad Shakeel

10-PhD-MT-03

Supervisor

Prof. Dr. Syed Tauseef Mohyud-Din Department of Mathematics HITEC University Taxila

iv


The thesis titled Modified )/( GG -Expansion Methods for Soliton Solutions of Nonlinear

Differential Equations

submitted by

Muhammad Shakeel

Reg. No. 10-PhD-MT-03

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY IN MATHEMATICS

has been approved

External Examiner : _____________________________ Prof. Dr. Muhammad Ozair Ahmad

Department of Mathematics University of Engineering & Technology, Lahore

Supervisor : __________________________

Prof. Dr. Syed Tauseef Mohyud-Din Department of Mathematics HITEC University Taxila

Chairperson : __________________________ Prof. Dr. Muhammad Tahir Department of Mathematics HITEC University Taxila

Dean Faculty of Sciences: __________________________ Prof. Dr. Syed Tauseef Mohyud-Din Department of Mathematics HITEC University Taxila

v

Declaration

I, Muhammad Shakeel, 10-PhD-MT-03 hereby affirm that I have produced the work

presented in this thesis, during the scheduled period of study. I also affirm that I have not

taken any material from any source except referred to and wherever due that amount of

plagiarism is within acceptable range. If a violation of HEC rules on research has

occurred in this thesis, I shall be liable to punishable action under the plagiarism rules of

the HEC.

Date: ___________________

Signature of student:

Muhammad Shakeel

10-PhD-MT-03

vi

Certificate

It is certified that Mr. Muhammad Shakeel, registration number 10-PhD-MT-03 has

carried out all the work related to this thesis under my supervision at the Department of

Mathematics, HITEC University Taxila, and the work fulfills the requirement for award

of PhD degree.

Date: ____________________

Supervisor:

_____________________________ Prof. Dr. Syed Tauseef Mohyud-Din Department of Mathematics HITEC University Taxila

Chairperson:

_____________________________

Prof. Dr. Muhammad Tahir Department of Mathematics HITEC University Taxila

vii

Dedicated

to

My beloved Parents & Wife for their prayers love and care

&

to

Ayesha & Hadi

viii

ACKNOWLEDGEMENTS

All praises be to Almighty Allah, The Creator of the entire universe and countless

Darood and Salaams to beloved Holy Prophet Hazrat Muhammad (Peace Be Upon Him),

for Whom this universe has been manifested. I thank to Almighty Allah, Whose blessings

bestowed upon me talented teachers, provided me sufficient opportunities and enabled

me to undertake and execute this research work.

My cordial appreciation goes to my affectionate, sincere, kind and most respected

supervisor Prof. Dr. Syed Tauseef Mohyud-Din, Dean Faculty of Sciences for so

patiently bearing and guiding me, invaluably suggesting and continuously encouraging

me with his precious contributions in completing this thesis. I am a l s o grateful for his

encouraging comments as well. If not for his invaluable advice and guidance, this thesis

would not have come to realization. He has been very kind in extending all possible help

to make this work a success. His ideologies and concepts have a remarkable impact on

my research contrivances. He genuinely facilitated me without which my objective would

not have been obtained. I have learnt a lot from his abilities.

I would also like to pay the most sincere gratitude to Vice Chancellor of HITEC

University, Brig (R) Qamar Zaman for the provision of research oriented and

encouraging environment coupled with the appropriate facilities. His sincere efforts to

streamline the system can be considered a role model. His personal interest in the subject

proved to be the real driving force and the essential inspiring factor behind the success of

this work. It was his dynamic and proactive approach that enabled me to learn

mathematics and do the research under the guidance of eminent mathematicians of

national and international repute.

I would like to pay sincere thanks to Chairman Department of Mathematics, Professor

Dr. Muhammad Tahir and all the faculty members for their moral support, kind

comments and confidence boosting attitude. Special thanks to Dr. Asif Waheed being

helpful, especially in early stages in Maple programming collaboration with my

supervisor and Dr. Saira Zainab,in compiling of the thesis. They always guided me

sincerely and honestly throughout my course work as well as research work. Sincere

thanks to Mr. Syed Adeel Akhtar Shah (Assistant Professor) and Ms. Nashrah Usmani for

ix

editing and improving the language.

Finally, I am also very thankful to Prof. Dr. S. Kamran Afaq, Chairman Department of

Mechanical Engineering and Prof. Dr. Jameel Ahmed Chairman Department of Electrical

Engineering, for providing well equipped computer lab facility for my research work.

Last but not the least; I would also pay sincere thanks to my parents and my wife, though

their long lasting prayers have opened new horizons for my success. I pay regards to my

caring and loving children Ayesha Shakeel and Abdul Hadi, my sister, my brothers and

my friends, whose sincere prayers, best wishes always make me courageous and daring

throughout my life.

Muhammad Shakeel

10-PhD-MT-03

x

ABSTRACT

Modified )/( GG -Expansion Methods for Soliton Solutions of

Nonlinear Differential Equations

Solitons play a pivotal role in many scientific and engineering phenomena. Solitons are a

special kind of nonlinear waves that are able to maintain their shape along the

promulgation. From the last four decades, the rampant part of fundamental phenomenon

of soliton has successfully attracted the researchers from the physical and mathematical

sciences. Various branches of science like solid-state physics, plasma physics, particle

physics, biological systems, Bose-Einstein-condensation and nonlinear optics are

enjoying the benefits taken from soliton. Soliton research gives way to theoretical aspects

such as soliton existence, computation of soliton profiles and soliton stability theory by

using the tools of soliton dynamics and soliton interactions to applicative aspects.

The hub of this thesis is to search not only for the solitary solutions of nonlinear

differential equations but also for nonlinear fractional differential equations. This piece of

writing targets to give an intuitive grasp for; Further Improved )/( GG -expansion,

Extended Tanh-function, Improved )/( GG -expansion, Alternative )/( GG -expansion

with generalized Riccati equation, )/1,/( GGG -expansion and Novel )/( GG -

expansion methods. Moreover, we shall extend Novel )/( GG -expansion method to

nonlinear fractional partial differential equations arising in mathematical physics. For

multifarious applications, all the methods are glib to follow. In addition, these methods

give birth to several types of the solutions like hyperbolic function solutions,

trigonometric function solutions and rational solutions. The premeditated methods are

very efficient, reliable and accurate in handling a huge number of nonlinear differential

equations.

xi

TABLE OF CONTENTS

Introduction---------------------------------------------------------------------------------1

1. Preliminary Definitions-------------------------------------------------------------------7

1.1. Introduction-------------------------------------------------------------------------8

1.2. Some Basic Definitions of Solitary Wave Theory-----------------------------8

1.2.1. Soliton----------------------------------------------------------------------8

1.2.2. Solitary Waves-------------------------------------------------------------8

1.2.3. Traveling Wave------------------------------------------------------------9

1.2.4. Types of Traveling Wave Solutions------------------------------------9

1.2.4.1. Solitary Waves and Solitons----------------------------------9

1.2.4.2. Kink Waves---------------------------------------------------10

1.2.4.3. Periodic Waves-----------------------------------------------11

1.2.4.4. Peakons--------------------------------------------------------11

1.2.4.5. Cuspons--------------------------------------------------------12

1.2.4.6. Compactons---------------------------------------------------12

1.3. Fractional Derivative-------------------------------------------------------------13

1.3.1. Riemann-Liouville Fractional Derivative ----------------------------13

1.3.2. Modified Riemann-Liouville Fractional Derivative ----------------13

1.4 Integrability Tests-----------------------------------------------------------------15

1.4.1. Conservation Laws------------------------------------------------------15

1.4.2. Lax Pair-------------------------------------------------------------------16

1.4.3. Generalized Symmetries------------------------------------------------16

1.5. Analysis of the Methods---------------------------------------------------------19

2. Exact Solutions of Nonlinear Differential Equations by Further Improved

)/( GG -Expansion and Extended Tanh-Function Methods--------------------36

2.1. Introduction---------------------------------------------------------------------------------37

2.2. Numerical Examples----------------------------------------------------------------------38

xii

2.2.1. The (3 + 1)-Dimensional Potential-YTSF Equation-------------------------38

2.2.2. The (3 +1)-Dimensional Jimbo-Miwa Equation------------------------------46

2.2.3. The (2 + 1)-Dimensional CBS Equation---------------------------------------55

2.2.4. The Benjamin-Bona-Mahony (BBM) Equation------------------------------63

2.2.5. The Symmetric Regularized Long Wave Equation--------------------------71

3. Soliton Solutions of Nonlinear Evolution Equations by Improved )/( GG -

Expansion Method-----------------------------------------------------------------------79

3.1. Introduction---------------------------------------------------------------------------------80

3.2. Numerical Examples----------------------------------------------------------------------81

3.2.1. The Burgers Equation------------------------------------------------------------81

3.2.2. The Zakharov-Kuznetsov (ZK) Equation-------------------------------------83

3.2.3. The Boussinesq Equation---------------- ---------------------------------------87

3.2.4. The Coupled Higgs Equations--------------------------------------------------89

3.2.5. The Maccari System--------------------------------------------------------------93

3.2.6. The Fifth Order Caudrey-Dodd-Gibbon Equation---------------------------96

4. Applications of Alternative )/( GG -Expansion Method with Generalized

Riccati Equation------------------------------------------------------------------------102

4.1. Introduction-------------------------------------------------------------------------------103

4.2. Numerical Examples---------------------------------------------------------------------104

4.2.1. The (1 + 1)-Dimensional Kaup-Kupershmidt Equation--------------------104

4.2.2. The Sixth-Order Boussinesq Equation ---------------------------------------111

4.2.3. The Fifth Order CDGSK Equation-------------------------------------------116

4.2.4. The (3 + 1)-Dimensional Modified KdV-ZK Equation--------------------122

5. Traveling Wave Solutions of Nonlinear Partial Differential Equations by

)/1,/( GGG -Expansion Method--------------------------------------------------128

5.1. Introduction-------------------------------------------------------------------------------129


5.2.1. The Positive Gardner-KP Equation-------------------------------------------130

5.2.2. The (2 + 1)-Dimensional CBS Equation-------------------------------------138

5.2.3. The Modified Benjamin-Bona-Mahony Equation--------------------------142

xiii

6. Applications of Novel )/( GG -Expansion Method to Nonlinear Evolution

Equations---------------------------------------------------------------------------------150

6.1. Introduction-------------------------------------------------------------------------------151


6.2.1. The ZK-BBM Equation--------------------------------------------------------152

6.2.2. The Symmetric Regularized Long Wave Equation-------------------------168

6.2.3. The Boussinesq System--------------------------------------------------------184

6.2.4. The (3 + 1)-Dimensional Burgers Equations--------------------------------201

6.2.5. The (3 + 1)-Dimensional Modified KdV-ZK Equation--------------------208

7. Novel )/( GG -Expansion Method for Fractional Partial Differential

Equations---------------------------------------------------------------------------------219

7.1. Introduction-------------------------------------------------------------------------------220


7.2.1. The Time Fractional Simplified MCH Equation----------------------------221

7.2.2. The Time Fractional BBM-Burgers Equation-------------------------------234

7.2.3. The Space-Time Fractional SRLW Equation--------------------------------247

8. Conclusion--------------------------------------------------------------------------------263

9. References---------------------------------------------------------------------------------265

xiv

LIST OF TABLES

Table 1.1. The general representation of solutions of Eq. (1.11)--------------20

xv

LIST OF ABBRIVATIONS

YTSF Yu-Toda-Sasa-Fukuyama

CBS Calogero-Bogoyavlenskii-Schiff

BBM Benjamin-Bona-Mahony

SRLW Symmetric Regularized Long Wave

NLEEs Nonlinear Evolution Equations

ODE Ordinary Differential Equation

PDE Partial Differential Equation

KdV Korteweg-de Vries

ZK Zakharov-Kuznetsov

CDG Caudrey-Dodd-Gibbon

CDGSK Caudrey-Dodd-Gibbon-Sawada-Kotera

MKdV Modified Korteweg-de Vries

KP Kadomtsev-Petviashvili

MBBM Modified Benjamin-Bona-Mahony

MCH Modified Camassa-Holm

xvi

LIST OF FIGURES

Fig. 1.1: General representation of Solitary Waves----------------------------------------------9

Fig. 1.2: Traveling Waves---------------------------------------------------------------------------9

Fig. 1.3: General representation of Solitons-----------------------------------------------------10

Fig. 1.4: General representation of Kink solution ---------------------------------------------10

Fig. 1.5: General representation of Periodic solution -----------------------------------------11

Fig. 1.6: General representation of Peakon solution-------------------------------------------11

Fig. 1.7: General representation of Cuspons----------------------------------------------------12

Fig. 1.8: General representation of Compacton ------------------------------------------------13

Fig. 2.1: Kink type solution of (2.37) for .0,0,2,1 0 zyp ---------------------------48

Fig. 2.2: Exact Kink solution of (2.38) for ,2,1 0 p .0,0 zy --------------------49

Fig. 2.3: Exact periodic traveling wave of (2.39) for ,0,1 0 p .0,0 zy ---------50

Fig. 2.4: Exact Kink solution of (2.43) for ,1,4/1 0 p .0,0 zy -------------------52

Fig. 2.5: Single soliton solution of (2.47) for ,3,2 0 p .0,0 zy -----------------53

Fig. 3.1: Singular soliton solution of (3.75) for .1,1,1 CBA ---------------------------99

Fig. 3.2: Bell-shaped 2sec h solitary wave solution of (3.76) for .1,1,1 CBA -------99

Fig. 3.3: Exact periodic wave solution of (3.77) for .2,1,1 CBA -------------------100

Fig. 3.4: Soliton solution of (3.78) for .2,1,1 CBA -----------------------------------100

Fig. 4.1: Solitons corresponding to solution 1u for .2,1,1 rqp ---------------------106

Fig. 4.2: Solitons corresponding to solution 20u for .2,1,3 rqp ------------------109

Fig. 4.3: Kink-type solution of 26u for .0,2,2 rqp -----------------------------------110

Fig. 4.4: Soliton solution of 27u for .0,3,0 rqp ----------------------------------110

Fig. 4.5: Solitons corresponding to solution 13u for .1,2,3 rqp -----------------114

Fig. 4.6: Singular soliton solution to solution 27u for .0,1,0 rqp ---------------116

Fig. 4.7: Solitons corresponding to solution 1u for .1,1,1 rqp ----------------------118

Fig. 4.9: Singular soliton solution of 14u for .1,2,3 rqp -------------------------------120

xvii

Fig. 4.10: Soliton solution of 27u for .0,2,0 rqp -------------------------------------122

Fig. 4.11: Exact periodic traveling wave solution of 1u for .9,2,1,1 rqp --123

Fig. 4.12: Soliton solution of 13u for .4,1,3,4 rqp -------------------------------125

Fig. 5.1: Soliton solution of 11u for .,1 xy ---------------------------------------------132



Fig. 5.4: Soliton solution of 23u for .,2 xy --------------------------------------------136

Fig. 5.5: Soliton solution of 23u for .,2 xy --------------------------------------------137

Fig. 5.6: Exact kink-type solution of 11u for .0,1,0,1 0 zk --------------------140

Fig. 5.7: Singular Kink solution of 21u for .0,1,0,1 0 zk ------------------------141

Fig. 5.8: Periodic traveling wave solution of 12u for ,0,1 0 .1,,1 wxzk ---142

Fig. 5.9: Soliton solution of 11u for .1 ----------------------------------------------------144

Fig. 5.10: Kink-type solution of 12u for .1 ----------------------------------------------145

Fig. 5.11: Soliton solution of 13u for .1 --------------------------------------------------147

Fig. 5.12: Periodic traveling wave solution of 23u for .2 --------------------------------147

Fig. 5.13: Periodic traveling wave solution of 14u for .1 --------------------------------148

Fig. 6.1: Graph of cuspon for 11u for .1,1,3,1,2,1,1 VbakCBA ---------155

Fig. 6.2: Singular soliton solution for 21u for .1,1,1,1,2,1,1 VbakCBA ---155

Fig. 6.3: Soliton solution for 91u for ,2,1,1 CBA .1,1,3,1 Vbak ------158

Fig. 6.4: Periodic solution for 121u for ,2,1,1 CBA .1,1,3,1 Vbak ----159

Fig. 6.5: Singular periodic solution for 131u for .1,1,3,2,2,1,1 VbakCBA ---160

Fig. 6.6: Singular periodic solution for 123u for .1,1,3,2,2,1,1 VbakCBA ---167

Fig. 6.7: Graph of cuspon for 11u for .1,1,2,1,1 VkCBA ----------------------171

Fig. 6.8: Singular soliton solution for 21u for ,2,1,1 CBA .1,1 Vk ------------171

Fig. 6.9: Soliton solution for 91u for ,2,1,1 CBA .1,1 Vk ----------------------174

xviii

Fig. 6.10: Periodic solution for 121u for ,2,1,1 CBA .1,1 Vk --------------------175

Fig. 6.11: Bell-shaped 2sech solution for 231u for .1,1,2,0,1 VkCBA -------178

Fig. 6.12: Periodic wave solution for 122u for ,2,1,1 CBA .1,2 Vk ----------181

Fig. 6.13: Singular periodic solution for 123u for .1,2,1,1 VCBA ------------------183

Fig. 6.14: Graph of cuspon for 11u for .1,1,1,1,2,1,1 VbakCBA --------187

Fig. 6.15: Soliton solution for 91u for ,2,1,1 CBA .1,3,1,1 Vbak -----190

Fig. 6.16: Periodic traveling wave solution for 121u for ,2,1,1 CBA ,2k

,1a .1,3 Vb ----------------------------------------------------------------191

Fig. 6.17: Bell-shaped 2sech solution for 231u for .1,3,1,1,2,0,1 VbakCBA --194

Fig. 6.18: Periodic traveling wave solution for 132u for ,2,1,1 CBA ,1k

.1,5,1 Vba ------------------------------------------------------------------197

Fig. 6.19: Singular periodic solution for 123u for .1,3,1,2,1,1 VbaCBA --199

Fig. 6.20: Kink-type solution of 11u for ,2,1,1 CBA ,1,1,1,1 cbak

.,10 zyx ----------------------------------------------------------------------203

Fig. 6.21: Periodic traveling wave solution for 121u for ,1,1,2,1,1 akCBA

,1,1 cb .,10 zyx -------------------------------------------------------204

Fig. 6.22: Soliton solution of 21u for ,0,9,2,5.0,1,3 ykCBA .0z ----210

Fig. 6.23: Exact periodic traveling wave solution of 121u for ,5.1,2 BA ,0C

.0,0,9,2 zyk ------------------------------------------------------------212

Fig. 6.24: Soliton solution of 131u for ,0,9,2,5.0,5.1,2 ykCBA .0z ---212

Fig. 6.25: Periodic traveling wave solution of 122u for ,5.1,2 BA ,0C ,9,1 k

.0,0 zy ----------------------------------------------------------------------------214

Fig. 6.26: Kink solution of 14u for ,2,5.1,1 CBA .0,0,9,1 zyk -------216

Fig. 6.27: Periodic traveling wave solution of 124u for ,2,1,1 CBA ,9,2 k

.0,0 zy -------------------------------------------------------------------------------217

Fig.7.1 (a - d): Exact kink solution of 11u for .1,1,1,1,2,1,1 LkCBA --224

xix

Fig.7.2 (a-d): Singular kink solution of 21u for .5,1,1,2,1,2,5 LkCBA ---225

Fig.7.3 (a -d): Periodic traveling wave solution of 122u for

.12,1,1,2,2,1,2 LkCBA --230

Fig.7.4 (a -d): Singular periodic traveling wave solution of 123u for ,2,2,1,1 kCBA

.9,1,1 L -----------------------------------------------------------------------------------232

Fig.7.5 (a-d): Singular soliton solution of 21u for .1,1,1,2,1,1 VLkCBA --237

Fig.7.6 (a-d): Bell-shaped 2sec h solution of 231u for .1,1,1,2,0,1 VLkCBA ----242

Fig.7.7 (a-d): Periodic traveling wave solution of 123u for ,2,2,1,2 kCBA

.1,1 VL ------------------------------------------------------------------------------------------246

Fig.7.8 (a, b): Bell-shaped 2sec h solution of 231u for .1,1,1,2,0,1 VLkCBA ---255

Fig.7.9 (a, b): Periodic traveling wave solution of 122u for ,1,2,1,2 kCBA

.1,1 VL -----------------------------------------------------------------------------------------258

Fig.7.10 (a, b): Singular periodic solution of 123u for .1,1,2,1,1 VLCBA ---261

1

Introduction

It is illustrious that nonlinear evolution equations are big wheel in expanding significant

dynamic behaviors in fluid physics, solid physics, elementary particle physics, biological

physics, chemistry and superconductor physics etc., the probing of the exact solutions to

nonlinear evolution equations can help us analyzing and understanding their manifold

dynamical properties. The presented piece of writing is quite a fascinating research topic

that how to solve riddle partial differential equations particularly cops grabbing solutions,

including traveling wave solutions and soliton solutions and the topic also mesmeric for

Mathematicians, Physicists and Dynamicists. The delving of solitary wave solutions for

the nonlinear equations performs a seminal part in many scientific and engineering areas

such as plasma physics, solid state physics, fluid mechanics, chemistry and many others.

Soliton was first unveiled in 1834 by the Scottish naval engineer John Scott Russell

[191], who heeded that a canal boat stopping suddenly gave rise to a solitary wave which

traveled down the canal for several miles, without breaking up or losing strength. Russell

clepes this phenomenon the “soliton.” Russell adduced an enthralling historical account

the weighty scientific observation as well as fetching article as:

“I was observing the motion of a boat which was rapidly drawn along a narrow channel

by a pair of horses, when the boat suddenly stopped-not so the mass of water in the

channel which it had put in motion; it accumulated round the prow of the vessel in a state

of violent agitation, then suddenly leaving it behind, rolled forward with great velocity,

assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap

of water, which continued its course along the channel apparently without change of form

or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate

of some eight or nine miles an hour, preserving its original figure some thirty feet long

and a foot to a foot and a half in height. Its height gradually diminished, and after a chase

of one or two miles I lost it in the windings of the channel. Such, in the month of August

1834, was my first chance interview with that singular and beautiful phenomenon which I

have called the Wave of Translation”.

Russell catching ideas got failed until 1965, when Zabusky and Kruskal [260] began to

use a finite difference approach to the study of KdV equation, which was procured by

Korteweg and de Vires [110]. Several analytical methods also led to a complete

2

apprehending of solitons, remarkably the inverse scattering transform proposed by

Gardner et al. [64] in 1967. The magnitude of Russell’s discovery was then fully

acknowledged. It was unmasked that mathematical and physical theory of “soliton”

paved the way to describe many phenomena in physics, electronics, and biology. A

historical account of the scientific development of solitons is found in [70, 76].

In the recent past nonlinear sciences have touched the summit of unparalleled

development and progress and in the similar context wide range of analytical, semi

analytical and numerical schemes have been developed to handle the complexity of

diversified physical problems. Most of the developed techniques have their limitations

tied with some inbuilt deficiencies including limited convergence, divergent results,

linearization, discretization, unrealistic assumptions, huge computational work and non-

compatibility with the versatility of physical problems. Exact traveling wave solutions of

KdV, mKdV, variant Boussinesq and Hirota-Satsuma equations were obtained recently

by Wang et al. [217], when they presented a reliable technique naming )/( GG -

expansion method. In this technique, second order linear ordinary differential equation

with constant coefficients is utilized as an auxiliary equation

.0)()()( GGG

Consequently, this work has been used to procure exact traveling wave solutions for the

nonlinear differential equations in mathematical physics [35, 39, 61, 65, 108, 125, 177,

186, 239]. Efficiency and convenience of the proposed algorithm is paid heed as it has

lesser computational work as compared to some existing schemes including, Exp-

Function [15, 37, 71, 136, 144, 170, 174, 180, 181, 253], Modified Exp-Function [211],

Tanh-Function [151-153, 208-210, 225, 226], F-Expansion [16, 33, 212, 215, 273], Sine-

Cosine [218, 224], Hirota Bilinear Transformation [84, 124, 143, 150, 277], Bäcklund

Transformation [88, 129, 147], Inverse Scattering Transform [19, 122], Adomian’s

Decomposition [1, 2, 5, 21, 22, 32, 47, 48, 50, 52, 94, 97, 99, 101, 102, 185, 203, 237],

Variational Iteration [3, 9, 13, 14, 53, 66, 78, 85, 176, 205, 234, 240, 242, 257],

Homotopy Perturbation [4, 8, 46, 73, 75, 109, 166, 256, 258], Homotopy Analysis [6, 10-

12, 49, 51, 118-121, 165, 251 ], Extended Transformed Rational Function [270], Sech-

Function [145] and other methods [29, 62, 63, 86, 133, 137, 138, 140, 148, 149, 241, 243,

246, 271].

3

It is worth mentioning that Ma [139] presented a novel class of exact explicit solutions to

the Korteweg-de Vires equation using its bilinear form. Such solutions contain

singularities of trigonometric and exponential wave functions. The functions used in the

Wronskian determinants were derived from eigen functions of the Schrödinger spectral

problem associated with complex eigen values and the resulting solutions were termed as

the complexion solutions. In a subsequent work, Ma and Maruno [142] developed the

complexiton solutions for Toda lattice equation through the Casoratian formulation and

hence obtained a set of coupled conditions which guaranteed Casorati determinants to be

the solution of Toda Lattice which consequently produced complexiton solutions.

Moreover, Ma and You [135] used variation of parameters for solving the involved non-

homogeneous partial differential equations and obtained solution formulas helpful in

constructing the existing solutions coupled with a number of other new solutions

including rational solutions, solitons, positions, negatons, breathers, complextions and

interaction solutions of the KdV equations. In fact, the exp-function method is restricted

to produce rational solutions in the form of transformed variables and such solutions can

be obtained easily by making use of other techniques including Wronskian and

Casoratian [134]. Recently, Ma et al. [141] presented a much more general idea to yield

exact solutions to nonlinear wave equations by searching for the so-called Frobenius

transformations. Also the solutions of the Riccati equation (1.16) and the generalized

Riccati equation (1.33) and (1.57) are presented in 1996 by Ma and Fuchssteiner [130]

much earlier than [60] and [278]. Moreover, if G satisfies (1.24), then )/( GGF

solves a Riccati equation .2CFBFAF Therefore, )/( GG -expansion method is

basically the expansion approach around an integrable equation including a Riccati

equation, which is presented systematically in [130]. Moreover, a lot of research can be

made in future on the basis of presented work. For example, if complexiton solutions can

be added by making an appropriate use of multiple exp-function method presented by Ma

et al. [144].

In1999, Homotopy Perturbation method was first proposed by J. H. He [74] and applied

to nonlinear equations. This technique has been employed to solve a large diversity of

linear and nonlinear problems. Yildirim [254, 256, 258] applied homotopy perturbation

method to solve both linear and nonlinear boundary value problems for the space-and

4

time-fractional telegraph equations, the fourth-order integro-differential equations, time-

dependent Cauchy reaction-diffusion equation. Yildirim and Momani [252] also exploit

this technique to obtain analytical solutions to a fractional oscillator, which is obtained

from the analogous equation of motion of a determined harmonic oscillator by replacing

the second-order time derivative by a fractional derivative of order with .20

Yildirim [255] again applied this technique to obtain approximate solutions of nonlinear

differential-difference equations.

In 1994, George Adomian [21] was the first to initiate and urbanized the Adomian

decomposition method and applied it to solve the frontier problems of physics. Later on,

substantial amount of research work has been empowered by using this method to a broad

class of linear and nonlinear ODEs, PDEs and integral equations as well. Kaya [103-105]

implemented this technique to find exact, approximate and solitary wave solutions of the

compound KdV-type and compound KdV-Burgers-type, the generalized Boussinesq type

and generalized modified Boussinesq equations. Kaya and Inan [93] applied ADM to find

the solitary wave solutions of the combined KdV and MKdV equations. El-Sayed and

Kaya [54, 55] implemented the Adomian decomposition technique to unearth the solitary

wave solutions and numerical solutions of the seventh-order Sawada-Kotera and a Lax’s

seventh-order KdV and explicit traveling wave solutions of Whitham-Broer-Kaup

equations. Kaya and El-Sayed [98, 95] used this method for acquiring the soliton-like and

solitary wave solutions of the potential Kadomtsev-Petviashvili and the generalized

Burgers-Fisher equations.

Fractional calculus theory [189] has been applied to extensive class of intricate problems

arising in physics, biology, mechanics, visco-elasticity, applied mathematics and

nonlinear dynamics. Local fractional calculus is naked as one of constructive tools to deal

with everywhere continuous but nowhere differentiable functions in different areas of

science and engineering [31, 44, 238, 245-247]. Due to numerous applications of local

fractional calculus in different areas, it becomes a hot topic now a day and is successfully

applied to different problems by a number of genius researchers. For example, Yang et al.

[244] investigate a family of local fractional differential operators on Cantor sets for the

heat-conduction equation and the damped wave equation in fractal strings. Yang et al.

[240] applied local fractional variational iteration method for solving the local fractional

5

Laplace equations with the different fractal conditions.

To escalate the assortment of applicability, many researchers extend the basic )/( GG -

expansion method such as, a generalization of the original )/( GG -expansion method

was prearranged by Zhang et al. [275] for the evolution equations with erratic

coefficients. To endeavor general traveling wave solutions, the genius researcher, Zhang

et al. [272] also offered an enhanced )/( GG -expansion method. A spanking new method

of the )/( GG -expansion scheme was offered by Zayed [265] in which, G gratifies the

Jacobi elliptical equation .242cbGaGG Zayed [268] again throws light on a

further another approach of the basic )/( GG -expansion method in which G is the

solution of the Riccati equation )()( 2 GBAG .

Akbar et al. [24] purveyed further new solutions of nonlinear evolution equations by

using the generalized and improved )/( GG -expansion scheme.

It is crystal clear that most of the physical phenomenon are nonlinear in nature and hence

appropriate solutions of the same are highly significant. A dire need to search for new

solutions of nonlinear differential equations cannot be neglected. Modifications of

)/( GG -expansion methods and subsequent extension for construction of the soliton

solutions of nonlinear physical problems [27, 30, 39, 43, 123, 167, 188, 194-202, 216,

219, 220, 231, 232, 239] are proved to be brain wave of this work. Efficiency, accuracy

and user friendly factors of proposed modifications in )/( GG -expansion method are

highly observed. Computational work and subsequent numerical results are fully

supportive of the reliability of suggested schemes. The work in thesis is organized as

follows:

In Chapter 1, brief introduction of soliton theory and some definitions of fractional

derivatives are presented. Analysis of modified techniques including further improved

)/( GG -expansion, extended tanh, improved )/( GG -expansion, alternative )/( GG -

expansion with generalized Riccati equation, )/1,/( GGG -expansion, a novel )/( GG -

expansion scheme for integer order partial differential equations and for fractional-order

nonlinear evolution equations have been discussed.

Chapter 2 contains some new work on traveling wave solution of nonlinear partial

differential equations [123, 195, 196, 220, 239] via further improved )/( GG -expansion

6

and extended tanh-methods.

Chapter 3 discusses the traveling wave solutions of nonlinear evolution equations [39,

167, 197, 269] by using improved )/( GG -expansion method. Moreover, physical and

mathematical aspects of various nonlinear equations are researched by appropriate

graphs.

In Chapter 4, the soliton solutions of nonlinear evolution equations [43, 194, 198, 232]

are constructed by using the alternative )/( GG -expansion method in which, generalized

Riccati equation is used as an auxiliary equation. Moreover, this method is greatly

capable of reducing the size of computational work as compared to other existing

methods.

Chapter 5 comprises the applications of )/1,/( GGG -expansion method to seek the

exact solutions of nonlinear partial differential equations [30, 188, 200]. Efficiency and

accuracy of suggested scheme is highlighted by graphical representation.

Chapter 6 witnesses a novel )/( GG -expansion method for nonlinear differential

equations [199, 216, 219, 231, 232]. Various types of solutions including cuspons, bell

shaped ,sec 2h soliton, singular soliton, singular kink, periodic traveling wave, singular

periodic traveling wave have been obtained accordingly.

Chapter 7 is devoted to the study of fractional differential equations. A novel )/( GG -

expansion method is proposed to seek the soliton solutions of nonlinear fractional partial

differential equations [27, 201, 202]. Physical properties of several nonlinear traveling

wave solutions are examined by graphs which are obtained for various values of .

Conclusions are framed in Chapter 8, which shows that the applied/proposed modified

versions are highly accurate, user friendly, efficient and involves lesser computational

work as compared to number of existing techniques. It is to be highlighted that the

suggested algorithms are extremely simple but highly effective and may be extended to

other nonlinear problems of diversified physical nature.

Chapter 9 comprises all the references, which have been cited in this work.

7

Chapter 1

Preliminary Definitions

8

1.1 . Introduction

This chapter provides the basic concepts of solitary waves and fractional calculus. This

chapter not only discusses different terms like solitons and solitary waves, but also

explains these terms as the type of travelling wave solutions. It also includes the analysis

of some modified versions of the well-known direct computational )/( GG - expansion

methods. This section is subdivided into three sections. In the first section some basic

definitions of solitary wave theory are given. The basic definitions of fractional calculus

are given in second section and the third and last section discusses analysis of modified

techniques including:

o Further improved )/( GG -expansion method,

o Extended tanh-method,

o Improved )/( GG -expansion method,

o Alternative )/( GG -expansion method with generalized Riccati equation,

o )/1,/( GGG -expansion Method

o A novel )/( GG -expansion method and

o A novel )/( GG -expansion method for fractional-order NLEEs,

1.2. Some Basic Definitions of Solitary Wave Theory

1.2.1. Soliton

A soliton [221] is a solitary wave which preserves its shape and velocity upon nonlinear

interaction with other solitary waves. A Soliton has following properties;

1. It is of permanent form.

2. It is localized within a region.

3. It can interact with other solitons and emerge from the collision unchanged,

except for a phase shift.

4. Soliton is caused by an elusive balance between nonlinear and dispersive effects.

1.2.2. Solitary Waves

These waves have soliton-like solutions of nonlinear evolution equations, describing

wave process in dispersive and dissipation media [221]. A sketch of these types of waves

is shown below;

9

Fig. 1.1: General representation of solitary waves

1.2.3. Traveling Wave

Traveling wave [221] is defined as a wave in which the medium is traveling in the

direction of propagation of wave. Traveling waves arises in the study of nonlinear

differential equations, where waves are represented by the form;

,, Vtxftxu

Fig. 1.2: Traveling waves

and V is the speed of wave propagation. For ,0V the wave travels in the positive x

direction, whereas the wave moves in the negative x direction for .0V

1.2.4. Types of Traveling Wave Solutions

Traveling wave solutions are of many types and they are of particular importance in

solitary wave theory. Here, six different types of traveling wave solutions are

described with their figures.

1.2.4.1. Solitary Waves and Solitons

asymptotically zero at large distances. The particular types of solitary waves are

solitons. The soliton solution is a spatially confined solution, hence,

0)( ,)(),( uuu as ., Vtx

10

Fig. 1.3 : General representation of solitons

The KdV equation is a pioneer model for analytic bell-shaped 2sech solitary wave

solutions.

1.2.4.2. Kink Waves

Kink waves [221] are traveling waves, which ascend or descend from one asymptotic

state to another. The Kink solution comes close to a constant at infinity. The standard

dissipative burgers equation,

,xxxt vuuuu

is a well known equation that gives kink solutions, where v is the viscosity coefficient.

The graphical representation of Kink wave is

Fig. 1.4 : Shows the graph of a kink solution .tanh1, txtxu

11

1.2.4.3. Periodic Waves

Periodic waves [221] are traveling waves that are periodic such as )cos( tx . The

standard wave equation xxtt uu gives periodic solutions. Graphical representation is

given below;

Fig. 1.5 : General representation of periodic solution for .)cos( tx

1.2.4.4. Peakons

Peakons [221] are the solitary wave solutions. In this case the traveling solutions are

smooth except for a peak at the corner of its crest. The Peakons are solutions retaining

their shape and speed after interacting. Peakons were investigated and classified as

periodic peakons and peakons with exponential decay. Some graphical representations of

peakons are given below;

Fig. 1.6 : General representation of Peakon .,tx

etxu

12

1.2.4.5. Cuspons

Solitons are also found in the form of Cuspons [221], where solution demonstrates cusps

at their peaks. Unlike Peakons, where the derivative at the climax fluctuates only by a

sign, the derivative at the hop of Cuspons deviates. It is imperative that the Soliton

solution together with its derivatives approaches zero as .0x

Fig. 1.7: General representation of Cuspon .exp, 6

1

txtxu

1.2.4.6. Compactons

It is a new class of solitons with compact spatial support such that each compacton is a

soliton confined to a finite core. Compactons [221] have some properties, which can be

described in one of the following ways:

1. They are of finite wavelength with compact support.

3. They are free of exponential tail.

4. They are described by the absence of inestimable wings.

5. They are robust soliton-like solutions.

Graphical representation is given below;

13

Fig. 1.8 : General representation of Compacton

1.3. Fractional Derivative

1.3.1. Riemann-Liouville Fractional Derivative

The Riemann-Liouville fractional derivative [82] of order α for a function f is defined by

x n

n

n

x dssfsxdx

d

nxfD

0

1.

1

(1.1)

Note: The main disadvantage of Riemann-Liouville fractional derivative is that the

fractional derivative of a constant does not vanish. If ,Cf then

.1

CxCDx (1.2)

1.3.2. Modified Riemann-Liouville Fractional Derivative

The modified Riemann-Liouville [90, 91] derivative is defined as;

.1,1,

,10,01

10

nnntf

dfftdt

d

tfDnn

t

t

(1.3)

The modified Riemann-Liouville derivative has properties

,

1

1

ttDt (1.4)

,tgDtftfDtgtgtfD ttt (1.5)

. tgtgfDtgDtgftgfD gtgt

(1.6)

14

Definition 1.1 [228] The Euler constant is given by

.5772156649.0ln1

Lim1

n

kn

nk

Definition 1.2 [228] For Cz \ ,...3,2,1,0 Euler’s Gamma function z is

defined as

,...3,2,1,0,0zRe/1

0zRe,0

1

zifzz

ifdtetz

tz

Theorem 1.1 [228] Euler's Gamma function satisfies the following properties:

1. For ,0zRe the first part of definition (1.2) is equal to

1

0

11 .lnz dtz

t

2. For Cz \ ,...3,2,1,0

.1zz z

3. For n

.!1 nn

4. For Cz \ ,...3,2,1,0

.z-1z- z

5. For 0zRe the following limit holds:

nzzzz

nn z

n

...21

!Limz

6. Let Cz \ .,...3,2,1,0 Then Euler’s Gamma function can be defined by

1

z ,1z

1

n

nz

en

zze

where is the Euler constant.

7. Euler’s Gamma function is analytic for all Cz \ .,...3,2,1,0

8. Euler’s Gamma function is not at all zero.

9. For all non-integer ,Cz

15

z

sinz-1z and

z

sinz-z

10. For half-integer urging, Nnn ,2/ has the meticulous form

,

2

!!22/

2/1

n

nn

where !!n is the dual factorial defined by

1,01

02.4.6...2.

01.3.5...2.

!!

n

evennnn

oddnnn

n

Theorem 1.2 [228] Let ,0a and suppose that f and g are analytic on ., haha

Then

xgJxfDk

xgDxfDk

xfgDk

ka

k

k

ka

ka

10

(1.6a)

for .2/haxa

1.4. Integrability Tests [79]

From the last thirty years, the study of integrability of nonlinear ordinary and partial

differential equations has been the theme of most significant research developments. For

looking into the integrability of systems of partial differential equations, different

techniques are used; like conserved densities, Lax pair and generalized symmetries.

1.4.1 Conservation Laws [221]

For any differential equation, a conservation law is a divergence term

,0

x

X

t

TXDTD xt (1.6b)

where the quantities T and X are called density and flux respectively and none of them

engrosses derivatives with respect to t, is called a conservation law. Integration of (1.6b)

yields

TdxP Constant, (1.6c)

provided X disappears at .

16

1.4.2 Lax Pair [221]

If a partial differential equation can be written in Lax pair form

PLLPdt

dL (1.6d)

where, L and P are one-parameter operators from a fixed Hilbert space to itself and are

one time differentiable. Then the given partial differential equation generally has an

unlimited number of first integrals, which assist to study it.

1.4.3 Generalized Symmetries [79]

A vector function of the form ,...,,,, xxx uuutxG is called a symmetry of the partial

differential equation

,,...,,, 2 mxxxt uuuuFu (1.6e)

if and only if equation (1.6e) remains unaffected under the substitution Guu

contained by the order . Hence,

GuFGuDt (1.6f)

must grasp up to order on any solution of (1.6g). Therefore, G must gratifies the

equation

,GuFGDt (1.6g)

where F is the Fréchet derivative of F.

Theorem 1.3 [87] Every Lie point, Lie-Bäcklund and non-local symmetry

x

uuxx

uuxXi

i

,...,,,...,, 11 (1.6h)

of differential equations

,,...,2,1,0,...,,, mu

FvvuuxF ss

(1.6i)

gives a conservation law for the system of differential equations containing Eqs. (1.6i)

and the adjoint equations

17

.,...,2,1,0,...,,,* mu

FvvuuxF ss

(1.6j)

Theorem 1.4 [28] For all n > 0, the KdV equation

0 xxxxn

t uuuuG (1.6k)

admits the multipliers

.1

1,,1 1

n

xx un

uu (1.6l)

The only supplementary acknowledged multipliers of the form

xxx uuutx ,,,,

are given by

,1if, nxut

.2if,3

1

3

1 3

nxuuut xx (1.6m)

Theorem 1.5 [28] For capricious wave speeds c(u), the only multipliers of form

xt uuutx ,,,,

divulge by the wave equation

0xxtt uucucuG (1.6n)

are given by

.,, xtxt xutuuu

Theorem 1.6 [28] For non-constant wave speed c(u), the wave equation (1.6n) admits

supplementary multipliers of form

xt uuutx ,,,, (1.6o)

iff 2

00

uucuc in terms of constants ., 00 uc

For these wave speeds the supplementary admitted multipliers are given by

18

.

,

,

0

02

02

uuxutu

uuxux

uutut

xt

x

t

(1.6p)

Theorem 1.7 [183] Consider a system of differential equations

0,...,,, 1 kuuux (1.6q)

captivating place from a variational theory, whose results are extrema of an exploit

integral J [x, u] with Lagrangian L[x, u]. If

uxauuuxaxx AAAiii ,,, (1.6r)

is a Lie point symmetry of J [x, u] and vuxWi ,, are defined by (1.6s).

.

,...

,1...

,

,1...

,,,

11

11

11

12

1

1

11

11

...

...

...

...

2

...

...

1

Ajji

Ajj

Ajji

jj

k

Aji

Aj

Ajji

jj

k

Ai

Ai

k

k

k

k

k

k

u

uxLv

u

uxLD

u

uxLv

u

uxLD

u

uxLvvuxW

(1.6s)

Then

1. the identity holds for capricious functions u(x);

uuxWuxLxuDuxLEu Aiiiu

AA ~,,,,~ (1.6t)

2. the limited conservation law

0~,, uuxWuLxuD Aiii (1.6u)

grasps for any result u(x) of Euler-Lagrange equations.

19

1.5 Analysis of the Methods

1.5.1. Further Improved )/( GG -Expansion Method

Consider the following nonlinear partial differential equation,

,0),,,,,,,,( zzyyttzyxt uuuuuuuuP (1.7)

where P is a polynomial in the unknown function ).,,,( tzyxuu The further improved

)/( GG -expansion method [23] proceeds in the following steps:

Step 1: The wave variable in four independent variables

)(),,,( utzyxu , ,tVzyx (1.8)

transforms (1.7) into an ordinary differential equation in the form

,0),,,,( uuuuH (1.9)

where primes stand for the ordinary derivatives with respect to .

Step 2: Integration of (1.9) yields constant(s) of integration, if possible. For simplicity,

the integration constant(s) may be neglected.

Step 3: Suppose that the solution of (1.9) can be expressed by means of a polynomial in

)/( GG as follows:

,)(0

in

ii

G

Gu

(1.10)

where i ( ,3,2,1i ) are constants provided 0n and )(= ξGG satisfies the

),()()()]([ 6422 GrGqGpG (1.11)

where qp, and r are arbitrary constants to be indomitable later.

20

Table 1.1. The solutions of (1.11) can be found in [259, 269]:

No )(ξG No )(ξG

1 2

1

22

2

))tanh(1(

)(sec

prpq

phqp or

2

1

22

2

))coth(1(

)(csc

prpq

phqp, 0p

6 2

12

)tan(2

)(sec

prpq

ppor

2

12

)cot(2

)(csc

prpq

pp, 0p , 0r

2 2

1

)2cosh(

2

qp

p

, 0p 0

7

2

1

22

2

644

rpqe

ep

p

p

, 0p

3 2

1

)2cos(

2

qp

p

or

2

1

)2sin(

2

qp

p

, 0p , 0 .

8 2

1

)2

1tanh(1

p

q

p or

2

1

)2

1coth(1

p

q

p0p , 0

4 2

1

)2sinh(

2

qp

p

, 0p , 0

9 2

1

4

2

641

p

p

erp

ep, 0p , 0q

5 2

1

2

)tanh(2

)(sec

prpq

php or

2

1

2

)coth(2

)(csc

prpq

php, 0p ,

0r

10

q

1 , 0p , 0r .

where rpq 42 .

21

Step 4: In (1.10), the confirmatory integer n can be found by applying the homogeneous

balancing principle between the leading order linear term(s) with the nonlinear term(s) of

the leading order in equation (1.9).

Step 5: Inserting (1.10) in (1.9) and utilizing (1.11) polynomials in )(ξGi and

)()( iGG are obtained. Equating to zero each coefficient of the polynomials capitulate

a system of equations for unidentified constants. The values of the unidentified constants

rqpαn ,,, and V can be found by solving the system of equations and putting the values

in equation (1.10), further novel and general exact solutions of equation (1.7) are

originated.

1.4.2 Extended Tanh Function Method

Consider the following nonlinear partial differential equation,

,0),,,,,,,,( zzyyttzyxt uuuuuuuuP (1.12)

where P is a polynomial in the unknown function ).,,,( tzyxuu

The extended tanh function method [60] works in the following steps:

Step 1: The wave variable,

)(),,,( utzyxu , ,tVzyx (1.13)

,0),,,,( uuuuH (1.14)

where primes stand for the ordinary derivative with respect to .

Step 2: If possible, integrate (1.14) term by term, one or more times. This yields

constant(s) of integration. For simplicity, the integration constant(s) can be set to zero.

Step 3: Suppose that the solution of (1.14) can be expressed by means of a polynomial in

)( as follows:

,)(0

n

iiu (1.15)

22

where i are constants and )( satisfy the following Riccati equation

)()( 2 A . (1.16)

The Riccati equation (1.16) has the subsequent solutions [60]:

(i) If 0A , then

)(tanh)( AA or ).(coth)( AA (1.17)

(ii) If 0A , then

)(tan)( AA or )(cot)( AA . (1.18)

(iii) If 0A , then

1

)( . (1.19)

Step 4: In (1.15), the affirmative integer n can be found by applying the homogeneous

balancing principle between the leading order linear term(s) with the nonlinear term(s) of


Step 5: Inserting (1.15) in (1.14) and utilizing (1.16), polynomials in )( i are obtained. Equating to zero each coefficient of the polynomials capitulate a system of algebraic

equations for unidentified constants. The values of the unidentified constants rqpαn ,,,

and V can be found by solving the system of algebraic equations and putting the values

in equation (1.10), further novel and general exact solutions of equation (1.7) are

originated.

1.4.3. Improved )/( GG -Expansion Method

Consider the following nonlinear partial differential equation

,0),,,,,,( xxtxttxt uuuuuuF (1.20)

where F is a polynomial in unknown function txuu , and its partial derivatives.

The improved )/( GG -expansion method [126] progresses in the following steps:

23

Step 1: The wave variable

),(),( utxu ,tcx (1.21)

transforms (1.20) into an ordinary differential equation in the form

.0),,,,( uuuuP (1.22)

Step 2: Assume the solution of (1.22) can be expressed in )/( GG form as follows:

,)(

)()(

0

iM

ii

G

Gu

(1.23)

where i are constants given 0M and G is the solution of the nonlinear ordinary

differential equation

22 GCGBGGAGG , (1.24)

in which primes stand for the ordinary derivatives with respect to . A, B and C are

arbitrary constants which are to be indomitable at a later stage.

Using the general solutions of (1.24), following four solutions of )(/)( GG are:

Case 1. When 0B and ,0442 ACAB then

)(

)(

G

G= ,

)1(2)1(22

22

1

22

21

ecec

ecec

CC

B (1.25)

Case 2. When 0B and ,0442 ACAB then

)(

)(

G

G= ,

2cos

2sin

2sin

2cos

)1(2)1(221

21

cic

cic

CC

B (1.26)

Case 3. When 0B and ,0)1( CA then

)(

)(

G

G= ,

cossin

sincos

)1(21

21

cc

cc

C (1.27)

24

Case 4. When 0B

and ,0)1( CA then

)(

)(

G

G= ,

coshsinh

sinhcosh

)1(21

21

cic

cic

C (1.28)

where tcx , c is wave velocity; A, B, C and 21 , cc are real parameters.

Step 3: In (1.23), the positive integer M is obtained by applying the balancing principle

between the highest order linear term(s) with the nonlinear term(s) of the highest order in

(1.22).

Step 4: Inserting (1.23) into (1.22) and utilizing (1.24) polynomials in iGG )/( are

obtained. A system of algebraic equations for M and c is obtained by equating to zero

the coefficient of apiece resulted polynomial. After solving the obtained system of

algebraic equations the values of the unidentified constants M and c can be found and

substituting these values in (1.23), more general exact traveling wave solutions can be

established.

Xinhua and Zhen [230] consider the perturbed periodic “good” Boussinesq equation

,2xxxxxxxxtt uuuu (1.28a)

with initial and boundary conditions

,0,,0, xxuxxu t (1.28b)

...,,2,1,0,;,2;, jtxutxu jj (1.28c)

where ε is a petite positive integer, µ is a unvarying, φ and ψ are 2π-periodic

C functions gratifying

,02

0

2

0 dxxdxx (1.28d)

and ju represents the derivative with respect to x.

25

Theorem 1.8 [230] The solution of the initial and boundary value problem (1.28e)-

(1.28g)

,21 2xxxxxxxxtt uuuu (1.28e)

,;0,,;0, xxuxxu t (1.28f)

...,2,1,0,;,2;, jtxutxu jj (1.28g)

uniquely exists, such that

00;,;, Mtxutxu x (1.28h)

for all x, t ≥ 0 and 0 < |ε| ≤ 0 . 00 M is a positive constant.

1.4.4. Alternative )/( GG -Expansion Method

Consider the nonlinear PDE in the form

.0),,,,,,( txxxttxt uuuuuuF (1.29)

Step 1: The wave variable,

),(),( utxu ,tVx (1.30)

renovates (1.29) into an ODE in the form

,0),,,,( uuuuQ (1.31)

where primes stand for the ordinary derivative with respect to .

Step 2: Amalgamate (1.31) if possible, succumbs integration constant(s).

Step 3: The solution of (1.31) is articulated in the form of )/( GG as follows

,)(0

in

ii

G

Gau

,0na (1.32)

and G satisfies

,2GqGprG (1.33)

26

where ia , qp, and r are arbitrary constants to be resolute later on.

Accordingly, different families of exact solutions of (1.33) are follows from [278]:

Family 1:

When 042 qrp and ),0or (0 qrqp the solutions of (1.33) are:

22

1 42

1tan4

2

1pqrpqrp

qG ,

,42

1cot4

2

1 222

pqrpqrp

qG

,4sec4tan42

1 2223 pqrpqrpqrp

qG

,4csc4cot42

1 2224 pqrpqrpqrp

qG

,44

1cot4

4

1tan42

4

1 2225

pqrpqrpqrp

qG

,

4sin

4cos4)4()(

2

12

22222

6

BpqrA

pqrpqrApqrBAp

qG

,

4sin

4cos4)4()(

2

12

22222

7

BpqrA

pqrpqrApqrBAp

qG

wherever A and B are non-zero real constants and gratifying the form 022 BA .

,

42

1cos4

2

1sin4

42

1cos2

222

2

8

pqrppqrpqr

pqrr

G

,

42

1cos)4(4

2

1sin

42

1sin2

222

2

9

pqrpqrpqrp

pqrr

G

27

,

)4(4cos4sin)4(

4cos2

2222

2

10

pqrpqrppqrpqr

pqrrG

,

)4(4cos)4(4sin

4sin2

2222

2

11

pqrpqrpqrpqrp

pqrrG

.

)4(44

1cos)4(24

4

1cos4

4

1sin2

44

1cos4

4

1sin4

222222

22

12

pqrpqrpqrpqrpqrp

pqrpqrr

G

Family 2:

When 042 qrp and ),0or (0 qrqp the solutions of (1.29) are:

qrpqrpp

qG 4

2

1tanh4

2

1 2213 ,

,42

1coth4

2

1 2214

qrpqrpp

qG

,4sec4tanh42

1 22215 qrphiqrpqrpp

qG

,4csc4coth42

1 22216 qrphqrpqrpp

qG

,44

1coth4

4

1tanh42

4

1 22217

qrpqrpqrpp

qG

,

4sinh

4cosh4)4()(

2

12

22222

18

BqrpA

qrpqrpAqrpBAp

qG

28

,

4sinh

4cosh4)4()(

2

12

22222

19

BqrpA

qrpqrpAqrpABp

qG

somewhere A and B are non-zero real constants and gratify the stipulation 022 AB .

,

42

1cosh4

2

1sinh4

42

1cosh2

222

2

20

qrppqrpqrp

qrpr

G

,

42

1sinh4

2

1cosh4

42

1sinh2

222

2

21

qrppqrprqrp

qrpr

G

,

44cosh4sinh4

4cosh2

2222

2

22

qrpiqrppqrpqrp

qrprG

,

44cosh44sinh

4sinh2

2222

2

23

qrpqrpqrpqrpp

qrprG

.

444

1cosh424

4

1cosh4

4

1sinh2

44

1cosh4

4

1sinh4

222222

22

24

qrpqrpqrpqrpqrpp

qrpqrpr

G

Family 3:

When 0r and ,0qp the solutions of (1.29) are:

,

sinhcosh25

ppdq

dpG

,sinhcosh

sinhcosh26

ppdq

pppG

where d is an arbitrary constant.

29

Family 4:

When 0q and ,0 pr the solution of (1.29) is:

,1

1

27dq

G

where 1d is an arbitrary constant.

Step 4: The positive integer n in equation (1.32) is obtained by the homogeneous

balancing principle among the leading order linear term(s) with the nonlinear term(s) of


Step 5: Inserting (1.32) in (1.31) and utilizing (1.33) polynomials in iG and iG are

obtained. Equating to zero every coefficient of the polynomials, succumbs a system of

algebraic equations for different unidentified constants. After solving the system, the

value of the unidentified constants can be found and putting the values in (1.32) precise

and different types of the traveling wave results of (1.29) are obtained.

1.4.5. )/1,/( GGG -Expansion Method

Suppose a nonlinear partial differential equation for ),,( tzxP is given by

,0),,,,,,( zxxxzxt uuuuuuP (1.34)

The )/1,/( GGG -expansion method [116] proceeds as follows:


),(PP ,tVzx (1.35)

where is nonzero constant, )(P the function of , converts (1.34) into an ordinary

differential equation as:

.0),,,,( uuuVuQ (1.36)

Step 2: Integrating (1.36) one or more times, yields constant(s) of integration. For

simplicity, the integration constant(s) may be set to zero.

Step 3: According to the projected method, the solution of (1.36) can be expressed as

30

,)( 1

10

nM

nn

nM

nnu (1.37)

where n and n are constants to be find out later on and )( and )( are given by

G

G)( ,

,

1)(

G (1.38)

which satisfy

.)()( GG (1.39)

The equations (1.38) and (1.39) yield

,2 . (1.40)

The three cases of the general solution of (1.39) are:

Case 1: When ,0 the general solution of (1.39) is

,coshsinh

BAG

and we have

,22

22

2

(1.41)

where .22 BA

Case 2: When ,0 the general solution of (1.39) is

,cossin

BAG

and we have

,22

22

2

(1.42)

where .22 BA

Case 3: When 0 the general solution of (1.39) is

31

and hence

,22

1 2

2

2

BA

(1.43)

where A and B are arbitrary constants.

Step 4: In (1.37), the positive integer M is consummate by balancing principle among

the linear term of utmost order with the utmost order nonlinear term.

Step 5: Inserting (1.37) in (1.36) along with (1.40) and (1.41) will yield a polynomial in

and , in which the degree of is not bigger than 1. Equating like powers of M

and M equal to zero, yields a set of algebraic equations for n Mn ,,3,2,1,0 and

n BAMn ,,0,,,,3,2,1 and .V

Step 6: Solve the obtained system in step 5 for n ,,,3,2,1,0 Mn n

BAMn ,,,,,,3,2,1 andV to determine the values of these constants. Putting the

values of these constants in (1.37), the hyperbolic type of solutions of (1.34) are obtained.

Step 7: Similarly, inserting (1.37) in (1.36) along with (1.40) and (1.42) for 0 (or

(1.40) and (1.43) for 0 ), the traveling wave solutions of (1.34) expressed by

trigonometric (or by rational) functions respectively are obtained.

Theorem 1.9 [128] If the functions gg and hh satisfy the equation

nkhhhg

u

uugfhd

dgv

k

k

n

1,......~

,~.,.,.~, 1

(1.43a)

where v is some constant, then

hvtxgtxu ,, (1.43b)

provides a traveling wave solution to the following equation

.1,,,...,,,,

k

x

uuuuutxfu

k

kkn

xt (1.43c)

32

1.4.6. Novel )/( GG -Expansion Method

Consider the nonlinear partial differential equation of the type

.0),,,,,,,,,( ttzzyyxxzyxt uuuuuuuuuq (1.44)

The steps of the novel GG / -expansion method [26] are as follows:


),(),,,( utzyxu ,tVzyx (1.45)

transforms (1.44) into an ordinary differential equation

,0),,,,( uuuuP (1.46)

which contains various ordinary derivatives of the unknown function u with respect to .

Step 2: Assume the solution of (1.46) is of the form

,)()(

m

mi

i

i ku (1.47)

where )(

)()(

G

G . (1.48)

Here m or m may be zero, but at the same time both can not be zero. i and k are

calculated later and G satisfy

.)( 22 GCGBGGAGG (1.49)

The Cole-Hopf renovation )(

)()(ln)(

G

GG

reduces (1.49) to the Riccati

equation:

.)()1()()( 2 CAB (1.50)

Equation (1.50) has individual twenty five solutions (see Zhu, [278] for details).

Step 3: The affirmative integer m in equation (1.47) is obtained by applying the

balancing principle between the linear term(s) and the nonlinear term(s) of the leading

order in equation (1.46).

Step 4: Polynomials in iGGk / and iGGk

/ are obtained by inserting (1.47)

together (1.48) and (1.49) in (1.48). A system of algebraic equations for i

33

),,,2,1,0( mi k and V is obtained by collecting each coefficient of the resulted

polynomials to zero.

Step 5: The value of the constants can be found by solving the obtained system. By

inserting the values of the constants together with the solutions of (1.49), various types of

the solutions of the nonlinear evolution equation (1.44) can be found.

Lemma 1.1 [106] Let VUT : be a bounded linear operator and let pu be a

convergent sequence in U with u as its limit, then pu converges to u in U implies that

puA converges to uA in V.

1.4.7. The Fractional Novel )/( GG -Expansion Method

Assume a fractional partial differential equation is given by

,10,0,...,,,, uDuDuuuS txtx (1.51)

where uDx and uDt

are modified Riemann-Liouville derivatives of txu , .

The fractional novel )/( GG -expansion method [202] works as follows:

Step 1: A fractional complex renovation was projected by Li et al. [115] which transform

the given fractional PDE into an ODE. The traveling wave variable

,, utxu

,11

0

t

Vx

L (1.52)

reduces (1.51) into an ordinary differential equation of integer order

,0,...,,, uuuuP (1.53)

where 0 and, VL are arbitrary constants with .0, VL

Step 2. Integrating (1.53) if possible, gives up constant(s) of integration which can be

premeditated later.

Step 3: Assume the trial solution of (1.53) is

,)()(

m

mi

i

i ku (1.54)

where

.)(

)()(

G

G (1.55)

34

Herein m or m may be zero, but at the same time both can not be zero. i and k are

calculated later and G gratifies

,)( 22 GCGBGGAGG (1.56)

where primes stand for the ordinary derivative with respect ; A , B and C are factual

constants.

The Cole-Hopf transformation )(/)()(ln)( GGG diminishes (1.56) in

the following Riccati equation:

.)()1()()( 2 CAB (1.57)

The solutions of the equation (1.57) can be found in [278].

Step 4: The confirmatory integer m can be found by applying the homogeneous

balancing principle between linear term(s) and the nonlinear term(s) of the leading order

in equation (1.53).

Step 5: Polynomials in iGGk / and iGGk

/ are obtained by inserting (1.54)

together (1.55) and (1.56) in (1.53). A system of algebraic equations for i

),,,2,1,0( mi k, L and V is obtained by collecting each coefficient of the resulted

polynomials to zero.

Step 6: The value of the constants can be found by solving the obtained system. By

inserting the values of the constants together with the solutions of (1.56), various types of

the solutions of (1.51) can be found.

Consider the Camassa-Holm equation

Rxtuuuuuuuuuu xxxxxxxxtxxxxtxxt ,0,2 (1.57a)

with initial condition

,1,0, 0 sRHxuxu s

where RHH ss denotes the Sobolev space and ,, are constant and .4/1,0

35

Theorem 1.10 [114] Assume sHu 0 for some constant .2

3s Then the problem

,,,

,,022

1

0

2212

Rxxutxu

Rxtu

uuIuuu xxxxt

(1.57b)

has a unique solution txu , in sHTC ;,0 for T > 0.

36

Chapter 2

Exact Solutions of Nonlinear Evolution Equations by

Further Improved )/( GG -Expansion and Extended Tanh-

Function Methods

37

2.1. Introduction

The rapid development of nonlinear sciences has provided an ample opportunity to

benefit from a wide range of reliable and efficient techniques which are of significant in

tackling physical problems even of highly multifarious nature. After the surveillance of

solitary phenomena by John Scott Russell [191] in 1834 and since the Korteweg-de Vries

(KdV) equation was solved by Gardner et al. [64] by the inverse scattering method,

finding exact solutions of nonlinear evolution equations (NLEEs) has turned out to be

one of the most thrilling and predominantly dynamic areas of research. The manifestation

of solitary wave solutions in nature is quite prevalent like Bell-shaped 2sec h solutions

and kink-shaped tanh solutions, model wave phenomena in elastic media, plasmas, solid

state physics, fluids etc. There are various schemes to unearth the exact solutions of

nonlinear partial differential equations. Apart from their physical importance, the closed-

form solutions of NLEEs, if accessible assist the numerical solvers in comparison, and

prove obliging in the immovability analysis. Soliton theory carries many methods to deal

with the problem of solitary wave solutions for NLEEs such as: Truncated Painleve

Expansion [112], Weierstrass Elliptic Function [111], Inverse Scattering [18], Jacobi

Elliptic Function Expansion [127], Generalized Riccati Equation [176], Tanh-Coth [227],

Wronskian and Casorati [124, 142, 146] methods.

Later on, Wang et al. [217] presented a reliable technique, which is called )/( GG -

expansion method and obtained exact traveling wave solutions for the nonlinear

evolution equations (NLEEs). In this method, following differential equation with

constant coefficients is used as supplementary equation

.0)()()( GGG

In this chapter, further improved )/( GG -expansion method and extended tanh-function

method are used for finding traveling wave solutions of nonlinear evolution equations

[123,195, 196, 220, 239]. In first method following differential equation is used as an

auxiliary equation

).()()()]([ 6422 GrGqGpG

38

This modified version gives more general and new solutions as compared to traditional

)/( GG -expansion method. This modified version is discussed by Akbar et al. [23] and

Zayed [266] to find the soliton solutions of nonlinear evolution equations. In extended

tanh-function method, we shall utilize the concept of traditional )/( GG -expansion

method coupled with following nonlinear Riccati equation as an auxiliary equation.

.)()( 2 A

This method has less computational work as compared to conventional )/( GG -

expansion method and further improved )/( GG -expansion method. In some cases,

solutions are identical with )/( GG -expansion method, extended tanh-function method

and further improved )/( GG -expansion method by selecting the suitable values of the

arbitrary constants. Fan [58], El-Wakil et al. [56, 57], Fan and Hon [59] and Abdou [17]

have successively applied extended tanh-function method to seek the traveling solutions

of nonlinear partial differential equations. It is important to observe that the proposed

algorithms give different solutions. These algorithms are more reliable and efficient as

compared to the fundamental )/( GG -expansion scheme.

2.2. Numerical Examples

In this section, to verify the efficiency of well known direct computational methods: further improved )/( GG -expansion method and extended tanh-function methods, few

intricate examples are given. The solutions obtained by the two techniques are useful,

efficient, and trustworthy. The presented techniques provide additional new and exact

traveling wave solutions than the other methods.

Example 2.2.1. The (3 + 1)-dimensional Potential-YTSF Equation

Consider the (3 + 1)-dimensional potential-YTSF equation [196] in the form

.03244 yyzxxzxxzxxxtx uuuuuuu (2.1)

The use of the wave variable

),(),,,( utzyxu ,tVzyx (2.2)

39

into the partial differential equation (2.1) readily gives ODE and upon integrating once

with integration constants are considered zero yields

0)(3)34( 2 uuuV , (2.3)

Taking into consideration the homogeneous balance between u and 2)(u in (2.3), we

deduce that 1n . Consequently, the solution of equation (2.3) becomes

,0,)( 101 GGu (2.4)

where 0 and 1 are constants to be indomitable later and )(G gratifies the following

nonlinear auxiliary equation,

).()()()]([ 6422 GrGqGpG (2.5)

Inserting (2.4) and (2.5) in (2.3), a system of algebraic equations in different powers of

G is obtained as follows:

,0443: 1112 pqVqqG

,0686323: 21111

221

4 qrVrprqG (2.6)

,04812: 121

6 qrqrG

.04812: 21

221

8 rrG

Solving the above system (2.6) and the following two sets of solution are obtained as:

Set 1.

,41 ,00 ,2 rpq ,4

3 pV (2.7)

Set 2.

,41 ,00 ,0q ,4

34 pV (2.8)

where ,0 p and r are arbitrary constants.

Therefore the trial solution for the set 1 becomes:

,4)( 0

G

Gu (2.9)

where tpzyx )4

3( .

40

The succeeding families of exact solutions due to step 3 of section 1.5.1 are:

Family 1. If ,0p the form of the solution is

2

1

22

2

))coth(1(

)(csc)(

prpq

phqpG or .

))coth(1(

)(csc)(

2

1

22

2

prpq

phqpG

In these cases the ratio is

rpqpprpprppq

rpprppprpppqp

G

G

2222

22

)cosh()sinh(2)(cosh2)(cosh

)(cosh2)cosh()sinh(2)cosh()sinh(

or

.


)(cosh2)cosh()sinh(2)cosh()sinh(2222

22

rpqpprpprppq

rpprppprpppqp

G

G

Because ,2 rpq consequently the solution is

02

2

2)2tanh(2)2(sec3

)2(sec84)(

pph

phppu

or .2)2tanh(2)2(sec3

)2(sec84)( 02

2

pph

phppu (2.10)

Family 2. If ,0p ,0r the form of the solution is

2

1

2

)tanh(2

)(sec)(

prpq

phpG or .

)coth(2

)(csc)(

2

1

2

prpq

phpG

Then the ratio is

)sinh(2)cosh()cosh(

)(cosh2)cosh()(sinh 2

prppqp

rpprpppqp

G

G

or

.)cosh(2)sinh()sinh(

)(cosh2)cosh()(sinh 2

prppqp

rpprpppqp

G

G


0)(tanh22)( pppu or .)(coth22)( 0 pppu (2.11)

41


2

1

2

)tan(2

)(sec)(

prpq

ppG or .

)cot(2

)(csc)(

2

1

2

prpq

ppG

Then the ratio is

)(cos(sin2)(cos

)(cos)sin()(cos2 2

pqprpp

ppqprprpp

G

G

or

.)(sin(cos2)(sin

)(cos)sin()(cos2 2

pqprpp

ppqprprpp

G

G


0)2sec()2(tan1

1)2(tan4)(

pp

ppu

or

.)2sec()2(tan1

1)2(tan4)( 0

pp

ppu

(2.12)

Family 4. If ,0p ,0 the type of the solution is

2

1

)2

1tanh(1)(

pq

pG or .)

2

1coth(1)(

2

1

pq

pG


)

2

1tanh(1

4p

p

G

G or .)

2

1coth(1

4

p

p

G

G

Subsequently the traveling wave solution is

0)2

1tanh(1)(

ppu or .)

2

1coth(1)( 0

ppu (2.13)

Family 5. If ,0p the type of the solution is

2

1

22

2

644)(

rpqe

epG

p

p

.

42

The ratio is

.

64168

6416224

24

rpqeqe

rpqep

G

Gpp

p


,16

4)( 02

2

p

p

erp

epu

(2.14)

where tpzyx )4

3( .

While, the trial solution for the set 2 becomes:

,4)( 0

G

Gu (2.15)

where tpzyx )4

34( .


Family 6. If ,0p ,0 the kind of the solution is

.)2cosh(

2)(

2

1

qp

pG

Because ,0q then .0r Then the ratio for the case is

).2tanh( ppG

G

Hence, the traveling wave solution is

.)2tanh(4)( 0 ppu (2.16)

Family 7. If ,0p ,0 the form of the solution is

.)2sinh(

2)(

2

1

qp

pG

43

Because ,0q then .0r The ratio for the case is

).2coth( ppG

G

Hence, the obtained solution in this case is

.)2coth(4)( 0 ppu (2.17)


2

1

)2cos(

2)(

qp

pG

or .

)2sin(

2)(

2

1

qp

pG

Because ,0q then .0r Thus the ratio is

)2tan( ppG

G

or ).2cot( pp

G

G

Consequently, the traveling wave solution is

0)2tan(4)( ppu or .)2cot(4)( 0 ppu (2.18)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or .

)coth(2

)(csc)(

2

1

2

prpq

phpG

Because ,0q the ratio in this case is

.)coth()tanh(2

1 ppp

G

G

Consequently, the solution is

.)coth()tanh(2)( 0 pppu (2.19)


2

1

2

)tan(2

)(sec)(

prpq

ppG or .

)cot(2

)(csc)(

2

1

2

prpq

ppG

44

Because in this case ,0q therefore the ratio is

.)tan()cot(2

1 ppp

G

G

.)tan()cot(2)( 0 pppu (2.20)

Family 11. If ,0p ,0q the form of the solution is

.641

)(2

1

4

2

p

p

erp

epG

Then the ratio is

,4

coth8

1

rrG

G

where 164 rp .

Hence, for the family 11 the solution is

.4

coth2

1)( 0

rru (2.21)


.1

)(

q

G

Then the ratio is

.1

G

G


,4

)( 0

u (2.22)

where tpzyx )4

34( .

45

Now, solving (2.1) by using extended tanh-function method. The solution of (2.1) is

,)()( 01 u (2.23)

.)()( 2 A (2.24)

Inserting (2.23) and (2.24) in (2.3), following polynomial equation in is obtained as

follows:

.0)3234(

)()3684()()63(

221

2111

21

2111

41

21

AAAVA

AAV

(2.25)

Equating to zero each coefficient of the polynomial equation (2.25), a system of

equations is obtained and after solving the system, following solution is obtained:

,21 ,00 ,4

3AV (2.26)

where 0 and A are arbitrary constants. Then according to step 3 of section 1.5.2, the

exact solutions of (2.1) are:

When 0A , the form of the solution is

0)tanh(2)( AAu or .)coth(2)( 0 AAu (2.27)

When ,0A the form of the solution is

0)tan(2)( AAu or .)cot(2)( 0 AAu (2.28)


,2

)( 0

u (2.29)

where

.)4

3( tAzyx

46

Remark 2.1: The results given in (2.27) and (2.16), (2.17) are alike, if we put ,4 pA

where .0p

Remark 2.2: The results given in (2.28) and (2.18) are alike, if we put ,4 pA

where .0p

Remark 2.3: Result in (2.29) is alike to (2.22).

Example 2.2.2. The (3 + 1)-Dimensional Jimbo-Miwa Equation

Consider the (3 + 1)-dimensional Jimbo-Miwa equation [123] in the form

.03233 zxtyyxxxxyyxxx uuuuuuu (2.30)

The Jimbo-Miwa equation is model equation for water waves of long wavelength with

weakly non-linear refurbishing forces and frequency dispersion [92]. The equation can

also be used to model waves in ferromagnetic media, as well as two-dimensional matter-

wave pulses in Bose-Einstein condensates.

Now solving (2.30) by further improved )/( GG -expansion method. Using (2.2) in

(2.30) that converts the given partial differential equation in an ordinary differential

equation and on integration with zero constant of integration yields:

.0)(3)32( 2 uuuV (2.31)

Taking into consideration the balancing principle between the u and 2)(u come out in

(2.33), we deduce that .1n Therefore, the trial solution can be written as

.)( 01

G

Gu (2.32)

Inserting (2.32) and (2.5) in (2.31), following algebraic equations in G are obtained as

follows:

,0432: 1112 pqqqVG

,0646323: 21111

221

4 qrVrprqG

,04812: 121

6 qrqrG (2.33)

.04812: 21

221

8 rrG

47

Solving the algebraic equations (2.33), the solutions are:

Set 1.

,2

32,2,,4 001 pVrpq (2.34)

Set 2.

,2

38,0,,4 001 pVq (2.35)

where 0 , p and r are arbitrary constants.

The form of the trial solution for the set 1 becomes

,4)( 0

G

Gu (2.36)

where tpzyx )2

32( .



2

1

22

2

))tanh(1(

)(sec)(

prpq

phqpG or .

))coth(1(

)(csc)(

2

1

22

2

prpq

phqpG

In these cases, the ratio is

rppprpprppq

rpprppprpppqp

G

G


))(cosh2)cosh()sinh(2)cosh()sinh((222

22

or

.)cosh()sinh(2)(cosh2)(cosh

))(cosh2)cosh()sinh(2)cosh()(sinh(2222

22

rpqpprpprppq

rpprppprpppqp

G

G

Because ,2 rpq subsequently, the traveling wave solution is

48

02

2

2)2tanh(2)2(sec

)2(sec84)(

pph

phppu

or .2)2tanh(2)2(sec3

)2(sec84)( 02

2

pph

phppu (2.37)

Fig. 2.1. Shows the form of the kink-type solution of (2.37) for .0,0,2,1 0 zyp


2

1

2

)tanh(2

)(sec)(

prpq

phpG or .

)coth(2

)(csc)(

2

1

2

prpq

phpG

Then the ratio is

))sinh(2)cosh(()cosh(

))(cosh2)cosh()(sinh( 2

prppqp

rpprpppqp

G

G

or .))cosh(2)sinh(()sinh(


prppqp

rpprpppqp

G

G

Because ,2 rpq subsequently, following traveling wave solution is

0)(tanh22)( pppu or .)(coth22)( 0 pppu (2.38)

49

Fig. 2.2. Shows the exact kink-type solution of (2.38) for .0,0,2,1 0 zyp


2

1

2

)tan(2

)(sec)(

prpq

ppG or .

)cot(2

)(csc)(

2

1

2

prpq

ppG

Then the ratio is

))(cos(sin2()(cos

))(cos)sin()(cos2( 2

pqprpp

ppqprprpp

G

G

or .))(sin(cos2()(sin


pqprpp

ppqprprpp

G

G

Because ,2 rpq consequently, obtained solution is

0)2sec()2(tan1

)1)2((tan4)(

pp

ppu

or .)2sec()2(tan1

)1)2((tan4)( 0

pp

ppu

(2.39)

50

Fig.2.3. Solution (2.39) represents the exact periodic traveling wave solution for ,0,1 0 p .0,0 zy


2

1

)2

1tanh(1)(

p

q

pG or .)

2

1coth(1)(

2

1

p

q

pG

Then the ratio is

)

2

1tanh(1

4p

p

G

G or .)

2

1coth(1

4

p

p

G

G

Subsequently, the solution is obtained as

0)2

1tanh(1)(

ppu or .)

2

1coth(1)( 0

ppu (2.40)


.64)4(

)(2

1

22

2

rpqe

epG

p

p

The ratio in this case is

.64168

)6416(224

24

rpqeqe

rpqep

G

Gpp

p

51

Because ,2 rpq consequently, the solution is

,16

4)( 02

2

p

p

erp

epu

(2.41)

where tpzyx )2

32( .

The form of the trial solution for the set 2 becomes:

,4)( 0

G

Gu (2.42)

where tpzyx )2

38( .

The successive families of exact solutions due to step 3 of section 1.5.1 are:


.)2cosh(

2)(

2

1

qp

pG

Because ,0q then .0r The ratio in this case is

).2tanh( ppG

G

Consequently, the wave solution is obtained in the form

.)2tanh(4)( 0 ppu (2.43)

52

Fig. 2.4. Shows the form of the exact kink solution of (2.43) for .0,0,1,4/1 0 zyp


.)2sinh(

2)(

2

1

qp

pG


)2coth( ppG

G

.

Consequently, the obtained solution is

.)2coth(4)( 0 ppu (2.44)

Family 8. If ,0p ,0 the solution of (2.5) has the form

2

1

)2cos(

2)(

qp

pG

or .

)2sin(

2)(

2

1

qp

pG

Since ,0q then .0r Thus the ratio is

)2tan( ppG

G

or ).2cot( pp

G

G


53

0)2tan(4)( ppu or .)2cot(4)( 0 ppu (2.45)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .

Because in this case ,0q the ratio is

)coth()tanh(2

1 ppp

G

G

.




2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

Because in this case ,0q the ratio is

)tan()cot(2

1 ppp

G

G

.


.)tan()cot(2)( 0 pppu (2.47)

Fig. 2.5. Solution (2.47) explains the single soliton for ,3,2 0 p .0,0 zy

54


2

1

4

2

641)(

p

p

erp

epG .

In this case the ratio is

rrG

G

4coth

8

1 ,

where 164 rp .

Consequently, the solution is obtained in the form

.4

coth2

1)( 0

rru (2.48)

Family 12. If ,0p ,0r then the form of the solution is

q

G1

)( .

In this case the ratio is

1

G

G.

Consequently, the traveling wave solution is obtained in the form

,4

)( 0

u (2.49)

where tpzyx )2

38( .

Now solving (2.30) by using the extended tanh-function method. The solution of (2.30)

can be represented as

,)()( 01 u (2.50)

where )( satisfy the Riccati equation,

55

).()( 2 A (2.51)

Substituting (2.50) and (2.51) in (2.31), following polynomial equation in is obtained

as follows:

0)3232(

)()3682()()63(

221

2111

21

2111

41

21

AAAVA

AAV

(2.52)

Equating to zero each coefficient of (2.52), a system of equations is obtained and solving

the system, following solution is attained:

,22

3,,2 001

AV (2.53)


exact solutions of (2.30) are:


0)tanh(2)( AAu or ,)coth(2)( 0 AAu (2.54)


0)tan(2)( AAu or ,)cot(2)( 0 AAu (2.55)


,2

)( 0

u (2.56)

where .22

3tAzyx

Example 2.2.3. The (2 + 1)-Dimensional CBS Equation

Consider the CBS equation [220] in the form

.024 zxxzxxzxxxtx uuuuuu (2.57)

The interface of a Riemann wave proliferating along the z-axis with a long wave along

the x-axis is described by Calogero-Bogoyavlenskii-Schiff equation. The (2 + 1)-

56

dimensional CBS equation occur in a non restricted form, but it can be written as a

system of differential equations and, in potential form, as a fourth-order partial

differential equation and admits a Lax representation and is integrable by the one-

dimensional inverse scattering transform. Bruzon et al. [41] find the symmetry reductions

and exact solutions of (2 + 1)-dimensional integrable CBS equation by using the classical

and nonclassical methods.

Now solving (2.57) by further improved )/( GG -expansion method. Using the traveling

Using the wave variable (1.8) that converts (2.57) into an ordinary differential equation

and upon integrating once with integration constant zero yields:

.0)(3 2 uuuV (2.58)

Applying the balancing principle between u and 2)(u in (2.58), we deduced that .1n

Thus,

.)( 10

G

Gu (2.59)

Substituting (2.59) and (2.5) in (2.58), following system of algebraic equation in G is

acquired:

,04: 112 pqqVG

,062323: 2111

221

4 qrVprqG

,04812: 12

16 qrqrG (2.60)

.04812: 21

221

8 rrG

Solving the above system of equations, two solution sets are obtained as follows:

Set 1.

,4,2,4, 100 pVrpq (2.61)

where 0 , p and r are arbitrary constants.

Set 2

,16,0,4, 100 pVq (2.62)

where 0 and p are arbitrary constants.

The form of the trial solution for the set 1 becomes:

57

G

Gu 4)( 0 , (2.63)

where .4 tpzx

The ensuing families of exact solutions due to step 3 of section 1.5.1 are:


2

1

22

2

))tanh(1(

)(sec)(

prpq

phqpG or

2

1

22

2

))coth(1(

)(csc)(

prpq

phqpG .


rppprpprppq

rpprppprpppqp

G

G


))(cosh2)cosh()sinh(2)cosh()sinh((222

22

or rpqpprpprppq

rpprppprpppqp

G

G

2222

22


))(cosh2)cosh()sinh(2)cosh()(sinh(

.

Because ,2 rpq consequently, the solution is

2)2tanh(2)2(sec

)2(sec84)(

2

2

0

pph

phppu

or 2)2tanh(2)2(sec3

)2(sec84)(

2

2

0

pph

phppu

. (2.64)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .

Then, the ratio is

))sinh(2)cosh(()cosh(


prppqp

rpprpppqp

G

G

or ))cosh(2)sinh(()sinh(


prppqp

rpprpppqp

G

G

.

Because ,2 rpq consequently, the wave solution is

58

)(tanh22)( 0 pppu or .)(coth22)( 0 pppu (2.65)


2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

Then, the ratio is

))(cos(sin2()(cos


pqprpp

ppqprprpp

G

G

or ))(sin(cos2()(sin


pqprpp

ppqprprpp

G

G

.

Because ,2 rpq consequently, the wave solution is

)2sec()2(tan1

)1)2((tan4)( 0

pp

ppu

or .)2sec()2(tan1

)1)2((tan4)( 0

pp

ppu

(2.66)


2

1

)2

1tanh(1)(

pq

pG or

2

1

)2

1coth(1)(

pq

pG .

Then, the ratio is

)2

1tanh(1

4p

p

G

G or

)2

1coth(1

4p

p

G

G.

Subsequently, the traveling wave solution is

)2

1tanh(1)( 0 ppu or .)

2

1coth(1)( 0

ppu (2.67)

59


2

1

22

2

64)4()(

rpqe

epG

p

p

. .


rpqeqe

rpqep

G

Gpp

p

64168

)6416(224

24

.

Because ,2 rpq consequently, the attained wave solution is

,16

4)(

2

2

0

p

p

erp

epu

(2.68)

where tpzx 4 .

For the set 2, the solution is:

,4)( 0

G

Gu (2.69)

where tpzx 16 .



2

1

)2cosh(

2)(

qp

pG

.

Because ,0q then ,0r so the ratio in this case becomes

)2tanh( ppG

G

.

Consequently, the obtained traveling wave solution is

).2tanh(4)( 0 ppu (2.70)

60

Family 7. If ,0p ,0 the solution of (2.5) has the form

2

1

)2sinh(

2)(

qp

pG

.

Because ,0q then .0r In this case the ratio is

)2coth( ppG

G

.


).2coth(4)( 0 ppu (2.71)

Family 8. If ,0p ,0 the solutions of (2.5) has the form

2

1

)2cos(

2)(

qp

pG

or

2

1

)2sin(

2)(

qp

pG

.


)2tan( ppG

G

or )2cot( pp

G

G

.


)2tan(4)( 0 ppu or ).2cot(4)( 0 ppu (2.72)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .

Because ,0q the ratio in this case is

)coth()tanh(2

1 ppp

G

G

.

.

61




2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

Because ,0q so in this case the ratio is

)tan()cot(2

1 ppp

G

G

.


.)tan()cot(2)( 0 pppu (2.74)


2

1

4

2

641)(

p

p

erp

epG .

For this case the ratio is

rrG

G

4coth

8

1 ,

where 164 rp .


.4

coth2

1)( 0

rru

(2.75)

Family 12. If ,0p ,0r then the form of the solution is

qG

1)( .

62

For this case the ratio becomes

1

G

G.


,4

)( 0

u (2.76)

where .16 tpzx

Now solving (2.57) by using extended tanh-function method. The solution of (2.57) can

be represented as

),()( 10 u (2.77)


).()( 2 A (2.78)

Substituting (2.77) and (2.78) in (2.58), following polynomial equation in is we

obtained as follows:

.0)32(

)()68()()63(22

12

11

22111

41

21

AAVA

AAV

(2.79)

A system of algebraic equations is obtained by equating to zero each coefficient of (2.79)

and after, the obtained solution is

,4,2, 100 AV (2.80)


exact solutions of (2.57) are


)tanh(2)( 0 AAu or ).coth(2)( 0 AAu (2.81)


)tan(2)( 0 AAu or ).cot(2)( 0 AAu (2.82)


63

,2

)( 0

u (2.83)

where .4 tAzx

Example 2.2.4. The Benjamin-Bona-Mahony (BBM) Equation

Consider the simplified BBM equation [239] in the form

.0 txxxxt uauuuu (2.84)

The Benjamin-Bona-Mahony equation (2.84) possesses precisely three sovereign and

non-trivial conservation laws [36]. If we put 1 vu in (2.84) with 1a , we get an

equivalent equation:

.xtxxt vvvv (2.84a)

The three conservation laws for the equation (2.84a) are:

,02

1 2

x

txt vvv (2.84b)

,03

1

2

1

2

1 322

x

tx

t

x vvvvv (2.84c)

.04

1

3

1 42223

x

txtxt

t

vvvvvv (2.84d)

Using the wave variable (1.8) in (2.84) that converts the given partial differential

equation into an ordinary differential equation and upon integrating once with integration

constant zero capitulates

.02

1)1( 2 aVuuuV (2.85)

Taking into consideration the balancing principle between the terms u and 2u in (2.85),

we deduce that .2n Therefore, the trial solution can be written as

.)( 012

G

G

G

Gu (2.86)

Substituting (2.86) and (2.5) in (2.85), following polynomial in G as follows:

64

.02

1

2

1

2

1

28

42

1

62

116

2

1

28242

1

21200

202

22220

1211011

121/

1213

22222

2120

2

22

222

212202

222

4

222

622

222

8

ppVpppV

pVGG

qVaqGGrVarGG

qqVpqVapqqqG

qVaprrVrrrprVaqG

rVaqrGrVarG

(2.87)

Proceeding as before and solving (2.87), the two solution sets are obtained as follows:

Set 1.

,116

482

ap

a ,01 ,

116

480

ap

ap ,0q ,

116

1

apV (2.88)

where p and a are arbitrary constant.

Set 2.

,116

482

ap

a ,01 ,

116

160

ap

ap ,0q ,

116

1

apV (2.89)

where p and a are arbitrary constant.

,116

48

116

48)(

ap

ap

G

G

ap

au (2.90)

where .116

1t

apx



2

1

)2cosh(

2)(

qp

pG

.

65


)2tanh( ppG

G

.

Consequently, the obtained wave solution is

).2(sec116

48)( 2 ph

ap

apu

(2.91)


2

1

)2sinh(

2)(

qp

pG

.


)2coth( ppG

G

.

Consequently, wave solution is

).2(csc116

48)( 2 ph

ap

apu

(2.92)


2

1

)2cos(

2)(

qp

pG

or

2

1

)2sin(

2)(

qp

pG

.


)2tan( ppG

G

or )2cot( pp

G

G

.

Consequently, the wave solution is

)2(sec116

48)( 2 p

ap

apu

or ).2(csc

116

48)( 2 p

ap

apu

(2.93)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .

66

Because ,0q and the ratio in this case is

)coth()tanh(2

1 ppp

G

G

or )(csc

2

2 phr

p

G

G

.


116

48)coth()tanh(

116

12)(

2

ap

appp

ap

apu

or

.116

48)(csc

116

12)( 4

2

ap

apph

apr

apu (2.94)


2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

Because 0q the ratio in this case is

)tan()cot(2

1 ppp

G

G

.

Consequently, the obtained solution is

.116

48)]tan()([cot

116

12)( 2

ap

appp

ap

apu (2.95)


2

1

)4exp(641

)2exp()(

prp

ppG .

Then the ratio is

rrG

G

4coth

8

1 ,

where 164 rp .

Consequently, the following traveling wave solution is

,116

48

4coth

1164

3)( 2

ap

ap

rapr

au

(2.96)

67

where .116

1t

apx

Similarly other solutions of the BBM equation can be found as well.

The trial solution for the set 2 becomes

,

116

16

116

48)(

ap

ap

G

G

ap

au

(2.97)

where .116

1t

apx

The succeeding families of exact solutions due to step 3 of section 1.5.1 are


.)2cosh(

2)(

2

1

qp

pG

Because ,0q .0r The ratio in this case is

)2tanh( ppG

G

.


.116

162tanh

116

48)( 2

ap

app

ap

apu (2.98)


2

1

)2sinh(

2)(

qp

pG

.

Because ,0q so .0r The ratio in this case is

).2coth( ppG

G


.116

16)2(coth

116

48)( 2

ap

app

ap

apu (2.99)

68


2

1

)2cos(

2)(

qp

pG

or

2

1

)2sin(

2)(

qp

pG

.


)2tan( ppG

G

or )2cot( pp

G

G

.


116

16)2(tan

116

48)( 2

ap

app

ap

apu

or .116

16)2(cot

116

48)( 2

ap

app

ap

apu (2.100)



2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .


116

16)]coth()tanh([

116

12)( 2

ap

appp

ap

apu

or

.116

16)(csc

116

12)( 4

2

ap

apph

apr

apu (2.101)


2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

69


)tan()cot(2

1 ppp

G

G

.


.116

16)]tan()([cot

116

12)( 2

ap

appp

ap

apu (2.102)


.)4exp(641

)2exp()(

2

1

prp

ppG

Then the ratio is

,4

coth8

1

rrG

G

where 164 rp .


,116

16

4coth

1164

3)( 2

ap

ap

rapr

au

(2.103)

where .116

1t

apx

Now solving (2.84) by using extended tanh-function method. The solution of the (2.85)

can be represented as

,)()()( 012

2 u (2.104)

where )( satisfy the Riccati equation

).()( 2 A (2.105)

Substituting (2.104) and (2.105) into (2.85), following polynomial equation in is

obtained as follows

70

.02

12)()2(

)()2

18(

)()2()()2

16(

202

20010111

22022

212

3211

4222

aVAVVaVA

VaVA

aVaV

(2.106)

Proceeding as before, the following two solution sets are obtained as;

Set 3.

,14

122

aA

a ,01 ,

14

40

aA

aA .

14

1

aAV (2.107)

Set 4.

,14

122

aA

a ,01 ,

14

120

aA

aA ,

14

1

aAV (2.108)

where A and a are arbitrary constants.

So, the exact solutions of (2.84) for the solution set 3 are:


14

4)(tanh

14

12)( 2

aA

aAA

aA

aAu

or .14

4)(coth

14

12)( 2

aA

aAA

aA

aAu (2.109)


14

4)(tan

14

12)( 2

aA

aAA

aA

aAu

or ,14

4)(cot

14

12)( 2

aA

aAA

aA

aAu (2.110)

where .14

1t

aAx

And, the exact solutions of (2.84) for the solution set 4 are:


14

12)(tanh

14

12)( 2

aA

aAA

aA

aAu

71

or .14

12)(coth

14

12)( 2

aA

aAA

aA

aAu (2.111)


14

12)(tan

14

12)( 2

aA

aAA

aA

aAu

or ,14

12)(cot

14

12)( 2

aA

aAA

aA

aAu (2.112)

where .14

1t

aAx

Example 2.2.5. The Symmetric Regularized Long Wave Equation

Consider the following SRLW equation [195],

,0 xxtttxxtxxtt uuuuuuu (2.113)

this equation occurs in numerous physical applications like ion sound waves in plasma.

This equation is symmetrical with respect to x and t.

Using the traveling wave variable (1.8) into (2.113) and integrating twice with integration

constants zeros yields

.02

11 222 uVuVuV (2.114)

Taking into consideration the balancing principle between the terms u and 2u in (2.114),

we deduce that .2n Thus, the trial solution is

.//)( 01

2

2 GGGGu (2.115)

Substituting (2.115) together with (2.5) into (2.114), the following polynomial equation

in G is obtained as follows:

72

.02

1

2

1

2

1

2

8)4

2

1()6

2

1

2

116(

)28()2

124(

21200

2202

2222

20

12110121

2112

21123

22

2022

22

212

222

2212

22220

222

22

24

222

26222

22

28

pVpVVVppVpV

pVVVGGqVqVGG

rVrVGGpqVqVqV

pqVqVqGqVrVr

qVrVprVrVprVG

qrVrqVGrVrVG

(2.116)

Proceeding as before the two solution sets are obtained;

Set 1.

.161

1,0,

161

48,0,

161

48012

piVq

p

pi

pi

(2.117)

Set 2.

.161

1,0,

161

16,0,

161

48012

piVq

p

pi

pi

(2.118)

The form of the trial solution for the set 1 becomes

,161

48/

161

48)(

2

p

piGG

piu

(2.119)

where ,161

1t

pix

while the form of the trial solution for the set 2 becomes

,161

16/

161

48)(

2

p

piGG

piu

(2.120)

where .161

1t

pix

According to the step 3 of section 1.5.1, the following subsequent families of exact

solutions are obtained as follows:

73


2

1

)2cosh(

2)(

qp

pG

.


)2tanh( ppG

G

.

Consequently, the traveling wave solutions are;

,161

482tanh

161

48)( 2

p

pip

p

piu

(2.121)

and for set 2, the solution is

.161

162tanh

161

48)( 2

p

pip

p

piu

(2.122)


2

1

)2sinh(

2)(

qp

pG

.


ppG

G2coth

.

Consequently, the traveling wave solution for the set 1 is

,161

482coth

161

48)( 2

p

pip

p

piu

and for the set 2 is

.161

162coth

161

48)( 2

p

pip

p

piu

(2.124)

74


2

1

)2cos(

2)(

qp

pG

or

2

1

)2sin(

2)(

qp

pG

.


)2tan( ppG

G

or )2cot( pp

G

G

.

Consequently, traveling wave solution for the set 1 is

p

pip

p

piu

161

482tan

161

48)( 2

or ,161

482cot

161

48)( 2

p

pip

p

piu

(2.125)


p

pip

p

piu

161

162tan

161

48)( 2

or .161

162cot

161

48)( 2

p

pip

p

piu

(2.126)


2

1

2

)tanh(2

)(sec)(

prpq

phpG or

2

1

2

)coth(2

)(csc)(

prpq

phpG .


)coth()tanh(2

1 ppp

G

G

or )(csc

2

2 phr

p

G

G

.

Consequently, the traveling wave solutions for the set 1 is

p

pipp

p

piu

161

48)coth()tanh(

161

12)(

2

75

or ,161

48)(csc

161

12)( 4

2

p

piph

pr

piu

(2.127)


p

pipp

p

piu

161

48)coth()tanh(

161

12)(

2

or .161

48)(csc

161

12)( 4

2

p

piph

pr

piu

(2.128)


2

1

2

)tan(2

)(sec)(

prpq

ppG or

2

1

2

)cot(2

)(csc)(

prpq

ppG .

Because ,0q so in this case the ratio becomes

)tan()cot(2

1 ppp

G

G

.


,161

48)tan()cot(

161

12)(

2

p

pipp

p

piu

(2.129)


.161

48)tan()cot

161

12)(

2

p

pipp

p

piu

(2.130)


2

1

)4exp(641

)2exp()(

prp

ppG .


rrG

G

4coth

8

1 ,

where 164 rp .

76


,161

48

4coth

1614

3)( 2

p

pi

rpr

piu

(2.131)


,161

48

4coth

1614

3)( 2

p

pi

rpr

piu

(2.132)

where .161

1t

pix

Now solving (2.113) by using extended tanh-function method. The solution of (2.113)

can be represented as:

,)()()( 012

2 u (2.133)


).()( 2 A (2.134)

Substituting (2.133) and (2.134) into (2.114), the following polynomial is obtained:

.02

12

)()2(

)()2

18(

)()2()()2

16(

202

220

20

1012

112

22022

212

2

3211

24222

2

VAVV

VVAV

VVVAV

VVVV

(2.135)

Proceeding as before, the two sets of solution are obtained as follows;

Set 3.

.41

1,

41

4,0,

41

12012

AiV

A

Ai

Ai

(2.136)

Set 4.

.41

1,

41

12,0,

41

12012

AiV

A

Ai

Ai

(2.137)

77

Thus, the exact solution of (2.113) for the solution set 3 have the following forms:

A

AiA

A

Aiu

41

4)(tanh

41

12)( 2

or .,41

4)(coth

41

12)( 2

A

AiA

A

Aiu

(2.138)

and for the solution set 4 for 0A

A

AiA

A

Aiu

41

12)(tanh

41

12)( 2

or .41

12)(coth

41

12)( 2

A

AiA

A

Aiu

(2.139)

For the solution set 3, when 0A the form of the solution is

A

AiA

A

Aiu

41

4)(tan

41

12)( 2

or .,41

12)(cot

41

12)( 2

AiA

A

Aiu

(2.140)

while for the solution set 4 for 0A

A

AiA

A

Aiu

41

12)(tan

41

12)( 2

or ,41

12)(cot

41

12)( 2

A

AiA

A

Aiu

(2.141)

where .41

1t

Aix

Discussion

Lately, the observation on the equivalence of results obtained by using extended tanh-

function method introduced by Fan [58] and original )/( GG -expansion method

introduced by Wang et al. [217] has been given by Parkes [187]. The same observation

has been pointed out by Kudryashov [113]. The results obtained by further improved

)/( GG -expansion method presented in this chapter have been compared with those

obtained by extended tanh-function method. From this study, it is observed that further

improved )/( GG -expansion method and extended tanh-function methods are not

78

equivalent. All the results obtained by extended tanh-function method are found by

further improved )/( GG -expansion method, when the parameters are given some

specific values. Similarly, some new solutions are also obtained.

79

Chapter 3

Soliton Solutions of Nonlinear Evolution Equations

by Improved )/( GG -Expansion Method

80

3.1. Introduction

The nonlinear evolution equations (NLEEs) are widely used as models to describe

complex physical phenomena in various fields of sciences, especially in fluid mechanics,

plasma waves and biology. One of the basic physical problems for those models is to

obtain their traveling wave solutions. Various techniques have been specifically used to

unearth different kinds of solutions of physical models described by nonlinear PDEs.

Stability and convergence must be considered in numerical methods [40, 67, 89] to avoid

divergent or inappropriate results. Several effective analytical and semi analytical

methods, including Homotopy Perturbation [72], Parameter Expansion [69], Extended

Tanh-Function [223] and Series Expansion [248, 249] methods have been developed

considerably to tackle nonlinear partial differential equations.

To dilate the possibility of applications, effectiveness and trustworthiness of the )/( GG -

expansion method, several researchers studied various nonlinear PDEs to generate

traveling wave solutions via improved )/( GG -expansion method and can be found in

[24, 262, 263]. Shehata [204] also modified the basic )/( GG -expansion method to

derive traveling wave solutions for nonlinear Schrodinger equation and the cubic-quintic

Ginzburg Landau equation. Zhang [274] explored a new application of this method to

some special nonlinear evolution equations, the balance numbers of which are not

positive integers. Recently, Liu et al. [126] proposed a new modification of traditional

)/( GG -expansion method to seek the traveling wave solution of NLEEs. In this

modified version the second order nonlinear ordinary differential equation

,22 GCGBGGAGG

is used as an auxiliary equation.

The transformed rational function method used by Ma [130, 132] and the basic )/( GG -

expansion method introduced by Wang et al. [217] have a common idea, i.e. to put the

given NLEE into the corresponding ordinary differential equation (ODE), then the ODE

can be transformed into a system of algebraic polynomials with the determining

constants. The exact traveling wave and rational solutions of the NLEE can be obtained

by the solutions of the ODE, however, the linear superposition principle [131] and

multiple exp-function method [144] is applierd to get the N-soliton and N-wave solution

81

of the PDE. Furthermore, improved )/( GG -expansion method handles NLEEs in a

direct manner with no initial/boundary conditions requirement.

In this chapter, an improved )/( GG -expansion method has been utilized to construct the

traveling wave solutions of nonlinear evolution equations [39, 167, 197].


In this section, some examples to elaborate the efficiency of the well known technique,

improved )/( GG -expansion method is discussed. The proposed method is effective,

reliable and credible. Several examples are given to illustrate the reliability and

performance of the projected scheme.

Example 3.2.1. The Burgers Equation

Consider the following Burgers equation [197] in the following form

.0 xxxt uuuu (3.1)

The applications of the Burgers equation can be found in applied mathematics, such as

modelling of fluid dynamics, turbulence, shock wave formation and traffic flow.

Making use of the wave transformation (1.8) into (3.1) gives

.0 uuuuV (3.2)

After integrating (3.2) and setting the integration constant as zero yields

.02

1 2 uuuV (3.3)

Considering the homogeneous balance between u and ,2u we deduce that 1M .

Therefore, according to step 2 of section 1.4.3, the trial solution (1.23) turns out to be

,0,)(

)()( 110

G

Gu (3.4)

where 0 and 1 are constants and need to be determined. Substituting (3.4) along with

(1.24) into (3.3), a system of equations in iGG )/( is obtained. A set of algebraic

82

equations with respect to unknowns is obtained by setting each coefficient of

2,1,0,)/( iGG i equal to zero

.02

1:/

,0:/

,02

1:/

2111

2

1101

1

2010

0

CGG

BVGG

AVGG

(3.5)

Solving the system of equations (3.5); two sets of solution are obtained as follows:

Set 1.

.12,44,44 12

02 CACABBACABV (3.6)

Set 2.

.12,44,44 12

02 CACABBACABV (3.7)

Substituting (3.6) and (3.7) into (3.4) and according to (1.25)- (1.28), three types of the

solutions of (3.1) are obtained as

Case 1. When 0B

and ,04421 ACAB then the exponential function

solutions can be found as

.1),(

22

21

22

21

111

11

tVxtVx

tVxtVx

ecec

ecectxu (3.8)

Case 2. When 0B

and ,04421 ACAB then the triangular function

solutions will be

.

2cos

2sin

2sin

2cos

1),(1

2

1

1

1

2

1

1

1

tVxctVxic

tVxctVxictxu (3.9)

Case 3. If 0B and 0)1(2 CA , then the triangular function solutions are

.cossin

sincos1),(

2221

22212

tVxctVxc

tVxctVxctxu (3.10)

83

Case 4. Again, if 0B and ,0)1(2 CA then the hyperbolic function solutions are:

,coshsinh

sinhcosh1),(

2221

22212

tVxctVxci

tVxctVxcitxu (3.11)

where A, B, C and 21 , cc are real parameters.

In particular, if ,21 cc (3.8) becomes,

.2

tanh1),( 11

tVxtxu (3.12)

Again, at ,21 cc (3.8) becomes

.2

coth1),( 11

tVxtxu (3.13)

Similarly, other solutions from cases 2-4 are obtained as well by giving particular values

to 1c and .2c

Example 3.2.2. The Zakharov-Kuznetsov (ZK) Equation

Consider the ZK equation [197] in two spatial and one time, dimensions presented as

,0xyyxxxt uubuuau (3.14)

where a and b are constants. Poole [190] showed that ZK equation (3.14) is not

completely integrable, as it does not pass any conventional integrability test like, the

inverse scattering transform and Painleve property. The three conservation laws for the

equation (3.14) are:

,02

2

yxyxxxt buDbuu

aDuD (3.14a)

,0223

2 2232

yxyyyxxyxxt ubuDuubuuubu

aDuD (3.14b)

84

.06

3

6

36

43

3

22

2

222

2222

223

yyxxxyxyy

yyyxxyyxyyxxxx

yyxxyxxx

xyxt

uuua

bubuD

uuuuuua

b

uua

buububuu

au

Duua

buD

(3.14c)

The equation (3.14) is converted into an ODE upon using the wave variable (1.8)

.022

2

ubua

uV (3.15)

Integrating (3.15) and neglecting integration constant yields

.022

2 ubua

uV (3.16)

Considering the homogeneous balance between u and ,2u we get ,22 MM i.e.

2M . Therefore, according to step 2 of section 1.4.3, the trial solution (1.23) turns out

to be

,0,)(

)(

)(

)()( 2

2

210

G

G

G

Gu (3.17)

where ,0 1 and 2 are constants and need to be dogged later. Inserting (3.17) along

with (1.24) into (3.16), a system of equations in iGG )/( is obtained as follows

85

.0242

11212:/

,04

820420:/

,0616

862

116:/

,0412

24:/

,0242

1:/

2222

22

4

2121

1212

3

12122

221

212

2

12

102

111

1

12

2200

0

CbabCbGG

Cba

CbBbbBCbGG

BbaVAb

BbBCbaACbGG

AbABb

aBbACbVGG

ABbAbaVGG

(3.18)

Proceeding as before the following two solution sets are obtained as follows;

Set 1.

.

124,

124

,124

,442

2

21

02

a

Cb

a

CbB

a

CbABACAbV

(3.19)

Set 2.

,

224,442

2

02

a

BACAbBACAbV

.

124,

1242

21a

Cb

a

CbB

(3.20)

Substituting (3.19) and (3.20) into (3.17) and according to equations (1.25)- (1.28), the

solutions for different cases are

Case 1. When 0B

and ,04421 ACAB then the exponential function

solutions can be found as

.

6),,(

2

22

21

22

2111

11

11

tVyxtVyx

tVyxtVyx

ecec

ecec

a

b

a

btyxu

(3.21)

Case 2. When 0B

and ,04421 ACAB the triangular function solution will be

86

.

2cos

2sin

2sin

2cos

6),,(

2

1

2

1

1

1

2

1

111

tVyxctVyxic

tVyxctVyxic

a

b

a

btyxu

(3.22)

Case 3. If 0B and 0)1(2 CA , then the triangular function solutions is

.cossin

sincos244),,(

2

2221

222122

tVyxctVyxc

tVyxctVyxc

a

b

a

btyxu

(3.23)

Case 4. Again, if 0B and 0)1(2 CA , the hyperbolic function solutions is

,4

coshsinh

sinhcosh24),,(

2

2

2221

22212

a

b

tVyxctVyxic

tVyxctVyxic

a

btyxu

(3.24)

where A, B, C and 21 , cc are real parameters and a, b are positive constants. equals to

2 or 6, so is . And if equals to 2, choose 2442 BACAbV and if equals to 6,

then V is 2442 BACAb . Similarly, if is equal to 2 or 6, V is

2442 BACAb or 2442 BACAb respectively.

As a special case, if 01 c and ,02 c then (3.23) becomes

.cot244

),,( 2222 tVyx

a

b

a

btyxu

(3.25)

And (3.24) becomes

.coth244

),,( 2222 tVyx

a

b

a

btyxu

(3.26)

Similarly, if 01 c and ,02 c then (3.23) becomes

.tan244

),,( 2222 tVyx

a

b

a

btyxu

(3.27)

And (3.24) becomes

.tanh244

),,( 2222 tVyx

a

b

a

btyxu

(3.28)

87

Similarly, the other solutions of (3.14) can be work out as well by giving particular

values to 1c and .2c

Example 3.2.3. The Boussinesq Equation

Now, consider the (1 + 1)-dimensional Boussinesq equation [197] in the form,

.02 xxxxxxxxtt uuuu (3.29)

This equation was proposed by Boussinesq for a model of nonlinear dispersive waves. It

explains the propagation of long waves in shallow water. Due to the existence of an

infinite number of conservation laws and N-soliton solutions, the Boussinesq equation

[80] is considered as completely integrable.

Equation (3.29) is converted into an ordinary differential equation upon using (1.21) and

integrating yields

,01 22 uuuV (3.30)

with constant of integrations are considered as zeros. Now,

balancing u and ,2u we

deduced that 2M . Therefore, the trial solution (1.23) becomes

,0,)(

)(

)(

)()( 2

2

210

G

G

G

Gu (3.31)

Inserting (3.31) along with (1.24) into (3.30), the following system of algebraic equations

in )/( GG is obtained as

.012266:/

,026410210:/

,0368

4338:/

,026

62:/

,023:/

2222

22

4

21211212

3

1202

221

21222

22

12

102

111121

12

22000

20

CCGG

CCBBCGG

BA

BBCACVGG

AAB

BACVGG

ABAVGG

(3.32)

88

Solving the above set of algebraic equations, two solution sets are obtained as follows;

Set 1.

.12,12,12,4142

2102 CCBCAABCAV (3.33)

Set 2.

.16,16,22,4142

212

02 CCBABACABCAV (3.34)

Substituting (3.33) and (3.34) in (3.31) and according to (1.25)- (1.28), the solutions of

(3.29) as:

Case 1. When 0B

and ,04421 ACAB the exponential function solution

will be

.

2

3

2),(

2

22

21

22

2111

11

11

tVxtVx

tVxtVx

ecec

ecectxu

(3.34)

Case 2. When 0B

and ,04421 ACAB then the triangular function solution

will be

.

2cos

2sin

2sin

2cos

2

3

2),(

2

12

11

1

2

1

111

tVxctVxic

tVxctVxictxu

(3.35)

Case 3. If 0B and 0)1(2 CA , then the triangular function solution is

.cossin

sincos62),(

2

2221

2221

22

tVxctVxc

tVxctVxctxu (3.36)

Case 4. Again, if 0B and 0)1(2 CA , then the hyperbolic function solution is

,coshsinh

sinhcosh62),(

2

2221

2221

22

tVxctVxic

tVxctVxictxu (3.37)

89

equals to 1 or 3, so is . And if equals to 1, then ABCAV 414 2 , if

equals to 3, then ABCAV 414 2 . Similarly, if is equal to 1 or 3, then

ABCAV 414 2 or ABCAV 414 2 respectively.

In particular, if ,21 cc then (3.34) becomes

,2

coth2

3

2),(

2

111

tVxtxu

(3.38)

and, if ,21 cc then (3.34) becomes

.2

tanh2

3

2),(

2

111

tVxtxu

(3.39)

Similarly, the other solutions of (3.29) can be found as well by giving particular values to

1c and 2c for cases (2-4).

Example 3.2.4. The Coupled Higgs Equations

Consider the following Higgs field equation [269] in the form

,0)(

,02

2

2

xxxxtt

xxtt

ubvv

uvuubauuu (3.40)

where for 0,0 ba the equation (3.40) is called the coupled Higgs field equation.

Here, by choosing ,1,0 ba (3.40) reduces to the following coupled Higgs field

equation

,0)(

,02

2

2

xxxxtt

xxtt

uvv

uvuuuu (3.40a)

By introducing a complex wave variable rtpxVtxWveUu i ,,,

that transforms (3.40a) into an ODE as follows

,0221

,02122

3222

UUUWVV

UWUUrpUV (3.41)

The second equation in (3.41) is integrating with constant of integration zero yields

(3.42)

90

Substituting (3.42) into the first equation of the system (3.41) yields

.0111 322224 UVUrpVUV (3.43)

Considering the homogeneous balance between U and ,3U we get 1M . Therefore, the

trial solution (1.23) becomes,

,0,)(

)()( 110

G

GU (3.44)

where 0 and 1 are constants and need to be determined. Substituting (3.44) along

with (1.24) into (3.43) yields a system of equations in .)/( iGG Setting each coefficient

of same power of )3,2,1,0(,)/( iGG i to zero, a system of algebraic equations with

respect to unknowns ,0 1 is obtained as follows

.024

4222:/

,0333333:/

,0223

223:/

,0

:/

21

41

14

143

122

113

1

3

1102

12

14

14

02

1

2

21

41

2211

41

2201

2

112

12

14

1201

221

30

210

2

14

02

022

0223

0

0

CVC

CcVVCGG

BCBVBCVBVGG

BVpVAAVBV

CArpCAVrVGG

cBAr

BAVprVpcGG

(3.45)

Solution of (3.45) gives

),1(44

12,44

1,44

)(21

2

22

12

22

02

22

C

ACAB

rpiB

ACAB

rpi

ACAB

rpV (3.46)

Substituting (3.46) in (3.44) and according to (1.25)- (1.28), the following solutions of

the coupled Higgs equation (3.40a) are obtained as follows:

Case 1. When 0B

and ,04421 ACAB the exponential function solutions

can be found as:

,

expexp

expexp1

121

2121

1

121

2121

1

1

22

11

cc

ccrpiU

91

,

expexp

expexp

2

2

121

2121

1

121

2121

111

cc

ccW

,exp

expexp

expexp1

121

2121

1

121

2121

1

1

22

11 rtpxicc

ccrpiu

.

expexp

expexp

2

2

121

2121

1

121

2121

111

cc

ccv

(3.47)

Case 2. When 0B

and ,04421 ACAB then the triangular function solutions

will be:

,

cossin

sincos1

121

2121

1

121

2121

1

1

22

12

cic

cicrpiU

,

cossin

sincos

2

2

121

2121

1

121

2121

112

cic

cicW

,expcossin

sincos1

121

2121

1

121

2121

1

1

22

12 rtpxicic

cicrpiu

.

cossin

sincos

2

2

121

2121

1

121

2121

112

cic

cicv

(3.48)

Case 3. If 0B and 0)1(2 CA , then the triangular function solutions are:

,cossin

sincos

412

2221

2221

2

22

23

cc

ccrpiU

,cossin

sincos2

2

2221

2221

23

cc

ccW

,expcossin

sincos

412

2221

2221

2

22

23 rtpxicc

ccrpiu

.cossin

sincos2

2

2221

2221

23

cc

ccv (3.49)

92

Case 4. Again, if 0B and 0)1(2 CA , then the hyperbolic function solutions

are:

,coshsinh

sinhcosh

412

2221

2221

2

22

24

cic

cicrpiU

,coshsinh

sinhcosh2

2

2221

2221

24

cic

cicW

,expcoshsinh

sinhcosh

412

2221

2221

2

22

24 rtpxicic

cicrpiu

,coshsinh

sinhcosh2

2

2221

2221

24

cic

cicv (3.50)

,12

tanh12,22

21

t

rpxcitxU

,12

tanh2,22

21

t

rpxtxW

,exp12

tanh12,22

21 rtpxit

rpxcitxu

.12

tanh2,22

21

t

rpxtxv (3.51)

Also, if 21 cc and ,4

1 and1,1 CBA then (3.47) becomes

93

,12

coth12,22

21

t

rpxcitxU

,12

coth2,22

21

t

rpxtxW

,exp12

coth12,22

21 rtpxit

rpxcitxu

.12

coth2,22

21

t

rpxtxv (3.52)

Similarly, other solutions of coupled Higgs equations (3.40a) for cases (2-4) can be found

as well.

Example 3.2.5. The Maccari System

Now, consider the Maccari system [167] in the form

.0)(

,02

xyt

xxt

uvv

uvuiu (3.53)

By introducing a wave variable ,,,, rtqypxVtyxWveUu i

that converts (3.53) into an ordinary differential equations as

.021

,02

UUWV

WUUprU (3.54)

Integrating the system (3.54) and neglecting the constant of integration yiels

.1 2UWV (3.55)

Inserting (3.55) into the first equation of (3.54) yields

.011 32 UUprVUV (3.56)

Considering the homogeneous balance between U and 3U in (3.56), we get 1M .

Therefore, the trial solution (1.23) becomes

.0,)(

)()( 110

G

GU (3.57)

94

Inserting (3.57) and (1.24) into (3.56), yields a system of equations in .)/( iGG Setting

each coefficient of same power of )3,2,1,0()/( iGG i to zero, the following system

of algebraic equations with respect to unknowns ,0 1 is obtained

.0242422:/

,033333:/

,023

222:/

,0:/

21111

31

211

3

112

1011

2

12

11201

211

211111

21

300

200110

20

CCVVCCVGG

BCVBCBBVGG

rBCA

BVVpACAVAVVrpGG

prVrBABAVVpGG

(3.58)

Solving the above system (3.58), the obtained solution is

),1(12,122

1,44

2

110

22 CVBVpCAABV (3.59)

Substituting (3.59) into (3.57) and according to (1.25)- (1.28), the solutions of (3.53) are

Case 1. When 0B

and ,04421 ACAB the exponential function solutions

can be found as

,

expexp

expexp12

2

1

121

2121

1

121

2121

1

11

cc

ccVU

,

expexp

expexp

2

2

121

2121

1

121

2121

111

cc

ccW

,442

1exp

expexp

expexp12

2

1

22

121

2121

1

121

2121

1

11

tpCAABqypxi

cc

ccVu

.

expexp

expexp

2

2

121

2121

1

121

2121

111

cc

ccv

(3.60)

95

Case 2. When 0B

and ,04421 ACAB then the triangular function solutions

will be

,

cossin

sincos12

2

1

121

2121

1

121

2121

1

12

cic

cicVU

,

cossin

sincos

2

2

121

2121

1

121

2121

112

cic

cicW

,442

1exp

cossin

sincos12

2

1

22

121

2121

1

121

2121

1

12

tpCAABqypxi

cic

cicVu

.

cossin

sincos

2

2

121

2121

1

121

2121

112

cic

cicv

(3.61)

Case 3. If 0B and 0)1(2 CA , then the triangular function solutions are

,cossin

sincos12

2221

2221

23

cc

ccVU

,cossin

sincos2

2

2221

2221

23

cc

ccW

,12exp

cossin

sincos12

2

2221

2221

23

tpCAqypxi

cc

ccVu

,cossin

sincos2

2

2221

2221

23

cc

ccv

(3.62)

Case 4. Again, if 0B and 0)1(2 CA , the hyperbolic function solutions are

,coshsinh

sinhcosh12

2221

2221

24

cic

cicVU

,coshsinh

sinhcosh2

2

2221

2221

24

cic

cicW

96

,12exp

coshsinh

sinhcosh12

2

2221

2221

24

tpCAqypxi

cic

cicVu

,coshsinh

sinhcosh2

2

2221

2221

24

cic

cicv (3.63)

where .442

1 22 tpCAAByx

In particular, if 21 cc and ,4

1 and1,1 CBA then (3.60) becomes:

,tanh12,,1 tVyxVtyxU

,tanh2,, 21 tVyxtyxW

,2exptanh12,, 221 tpqypxitVyxVtyxu

.tanh2, 21 tVyxtxv (3.64)

Also, if 21 cc and ,4

1 and1,1 CBA then (3.60) becomes:

,coth12,,1 tVyxVtyxU

,coth2,, 21 tVyxtyxW .

,2expcoth12,, 221 tpqypxiVtyxVtyxu

.coth2, 21 tVyxtxv (3.65)

Similarly, other solutions of the Maccari system (3.53) for cases (2-4) can be calculated

as well.

Example 3.2.6. The Fifth Order Caudrey-Dodd-Gibbon Equation

Finally, consider the fifth order CDG equation [39] in the form

.01803030 2 xxxxxxxxxxxxt uuuuuuuu (3.66)

This equation play an important part in numerous scientific applications such as solid

state physics, nonlinear optics, mathematical biology, nonlinear optics and quantum field

97

theory. The fifth order CDG equation [42] belongs to the class of completely integrable

equations and therefore, the conservation laws for the ranks 3, 4 and 6 are as follows

.57636002045763242

,144014405400108481224

252792184864

48963622

,12180

,2161803606

48367222

,2

623422

232

23

243

62

42324

33

2

222

22

42

3232

23425

2453624

42224

52

3224

2

222

23234233

323

uuuuuuuuuuT

uuuuuuuuuuu

uuuuuuuuuuu

uuuuuuuuuuuuuX

uuuuT

uuuuuuu

uuuuuuuuuuuuX

uuT

xxxxxx

xxxxxx

xxxxxxxxx

xxxxxxxxxx

xx

xxx

xxxxxxxxxx

x

(3.66a)

Using the wave variable (1.8) that carries (3.66) into an ODE as

.01803030 25 uuuuuuuuV (3.67)

Integrating (3.67) and with constant of integration zero yields

.06030 34 uuuuuV (3.68)

Now, balancing )4(u and ,3u we get 2M . Therefore, the trial solution (1.23) becomes

.0,)( 2

2

210

G

G

G

Gu (3.69)

Inserting (3.69) along with (1.18) into (3.68) yields a system of equations in

.)/( iGG Setting each coefficient of same power of )3,2,1,0()/( iGG i to zero, a

system of algebraic equations (which are omitted here for the sake of simplicity) with

respect to unknowns 0 , 1 , 2 and V is obtained and after solving the system yields the

following solution

.)1(,1,1

,168328162

210

2242222

CCBCA

AABBCAACBACV

(3.70)

98

Substituting (3.70) into (3.69) and according to (1.25)- (1.28), the solutions of the fifth

order Caudrey-Dodd-Gibbon (CDG) equation (3.66) are as follows:

Case 1. When 0B

and ,0442 ACAB then the exponential function solution

can be found as:

.14

),(

2

22

21

22

21

ecec

ecectxu (3.71)

Case 2. When 0B

and ,0442 ACAB then the triangular function solution

will be

.

2cos

2sin

2sin

2cos

14

),(

2

21

21

cic

cictxu (3.72)

Case 3. If 0B and 0)1( CA , then the triangular function solution will be

.cossin

sincos1),(

2

21

21

cc

cctxu (3.73)

Case 4. Again, if 0B and 0)1( CA , then the hyperbolic function solutions is

,coshsinh

sinhcosh1),(

2

21

21

cic

cictxu (3.74)

where .16832816 2242222 tAABBCAACBACx

In particular, if ,21 cc then (3.71) becomes:

.2

coth14

),( 2

txu (3.75)

99

Fig. 3.1. Represents the form of singular soliton solution of (3.75) for .1 CBA

Again, if ,21 cc then (3.71) becomes:

.2

tanh14

),( 2

txu (3.76)

Fig.3.2. Represents the bell-shaped 2sec h solitary traveling wave solution of (3.76) for ,1,1,1 CBA separated by infinite wings or infinite tails.

Similarly, if ,0,0 21 cc then (3.72) becomes:

.2

tan14

),( 2

txu (3.77)

100

Fig.3.3. Solution (3.77) shows the exact periodic traveling wave solution for .2,1,1 CBA

And, if ,0,0 21 cc then (3.72) becomes:

.2

cot14

),( 2

txu (3.78)

Fig. 3.4. Solution (3.78) explains the soliton for .2,1,1 CBA The particular types of

solitary waves are solitons. The soliton solution is a spatially confined solution, hence, 0)(),(),( uuu as , .tVx

Similarly, other solutions of the fifth order Caudrey-Dodd-Gibbon (CDG) equation (3.66)

for cases (2-4) can be calculated as well.

101

Discussion

In this chapter, an improved )/( GG -expansion method has been used to obtain new

traveling wave solutions of various nonlinear evolution equations. On contrary to basic

)/( GG -expansion method, the benefits of the presented scheme is that all the nonlinear

partial differential equations, which can be solved by basic )/( GG -expansion method,

are easily solved by improved )/( GG -expansion method. Many new and exact

traveling wave solutions have been found. The analysis and computations of the

improved )/( GG method are absolutely resourceful in finding the exact solutions.

102

Chapter 4

Applications of Alternative )/( GG -Expansion Method

with Generalized Riccati Equation

103

4.1. Introduction

The nonlinear differential equations take place in diversified physical phenomena in

science and engineering and many enthusiastic researchers explored the exact solutions

of nonlinear PDEs. There is a widespread category of non-integrable nonlinear

differential equations, which are to some extent integrable because for some values of

parameters these equations become integrable. To gaze for the exact solutions of

nonlinear differential equations, many different methods are used including: Improved F-

Expansion [214], Projective Riccati Equation [233], Hirota [83], Exp-Function [175] and

Homogeneous Balance [276] methods.

Zhu [278] introduced the generalized Riccati equation mapping to solve the (2 + 1)-

dimensional Boiti-Leon-Pempinelle equation. In this generalized Riccati equation

mapping, he employed the following nonlinear ordinary differential equation with

constant coefficients as an auxiliary equation,

.2GqGprG

Li et al. [117] used the Riccati equation expansion method to solve the higher

dimensional NLEEs. Bekir and Cevikel [34] investigated nonlinear coupled equations in

mathematical physics by applying the tanh-coth method, combined with the Riccati

equation. Guo et al. [68] studied the diffusion-reaction and the mKdV equations with

variable coefficients via the extended Riccati equation mapping method. Salas [192] also

used the projective Riccati equation method to obtain some exact solutions for

generalized Sawada-Kotera equation.

Recently, Akbar et al. [25] introduced a new approach of )/( GG -expansion method by

using first order nonlinear ordinary differential equation ,2GqGprG as an

auxiliary equation to construct the traveling wave solutions of fifth-order Caudrey-Dodd-

Gibbon equation. In the subsequent work Zayed [268] also used this method to find the

exact solutions of nonlinear PDEs.

This chapter includes the applications of alternative )/( GG -expansion method in which

generalized Riccati equation is used as an auxiliary equation to construct the soliton

solutions of nonlinear evolution equations [43, 194, 198, 232]. In this method, the

solutions are expressed in terms of trigonometric, hyperbolic, exponential, rational

104

trigonometric, rational hyperbolic and rational exponential functions. This modified

version is very simple and straightforward and gives more general as well as new

solutions as compared to basic )/( GG -expansion method.


In this section, some examples are discussed to elaborate the effectiveness of the well

known technique, alternative )/( GG -expansion method with generalized Riccati

equation. Numerical consequences together with the graphical depiction evidently

divulge the complete trustworthiness and competence of the projected algorithm.

Example 4.2.1. The (1 + 1)-Dimensional Kaup-Kupershmidt Equation

Consider the (1 + 1)-dimensional Kaup-Kupershmidt equation [194] in the form,


The fifth-order KK equation (4.1), is very important nonlinear partial differential

equation, belongs to the ranking of integrable equations with Lax pair representation. The

KK equation has properties similar to the well-known KdV hierarchy. Physically it

occurs in the promulgation of shallow water and magneto-sound promulgation in

plasmas.

The (1 + 1)-dimensional Kaup-Kupershmidt equation (4.1) [213] divulge the Lax pair

representation as follows:

.21047545309

,22235

3

xxxxxxxxxxx

xxx

uuuuuuuP

uuL

(4.1a)

,0202510 25 uuuuuuuuV (4.2)

where 5u indicates the fifth order derivative with respect to . Integrating (4.2) once

with respect to ,

105

,03

20)(

2

1510 324 uuuuuuVC (4.3)

where C is integration constant. According to step 3 of section 1.4.4, the solution of (4.3)

can be expressed by a polynomial in )/( GG as follows:

,0,...)(2

210

n

n

n aG

Ga

G

Ga

G

Gaau (4.4)

where ia are constants to be indomitable and G gratifies the generalized Riccati equation

.2GqGprG (4.5)

Considering the homogeneous balance between the highest order derivative and the

nonlinear terms in (4.3), we obtain .2n

Therefore, the trial solution takes the form:

.)(2

210

G

Ga

G

Gaau (4.6)

Using (4.5), equation (4.6) can be rewritten as:

.)()()( 212

110 GqGrpaGqGrpaau (4.7)

Substituting (4.7) into (4.3) the following polynomials in iG and iG , ),,3,2,1,0( ni

are obtained. Equating to zero each coefficient of the obtained polynomial, a system of

algebraic equations for rqpaaa ,,,,, 210 and V are obtained.

Solving the over-determined set of algebraic equations, following solution is attained:

.192

1

3

64

8

1122

,16716

,2

3,

2

3,2

8

6334222

2224

21

2

0

pqrqrpqrpC

qrqprp

Vap

arqp

a

(4.8)

Inserting (4.8) into (4.6) the following solution is obtained,

.2

3

2

32

8)(

22

G

G

G

Gprq

pu (4.9)

106

where .16716

2224

tqrqprp

x

Now, on the basis of the solutions of (4.5), the families of exact solutions of (4.1) are:

Family 1. When 042 qrp and ),0or (0 qrqp the solutions are:

,tan2

sec2

2

3

tan2

sec2

2

32

8

222222

1

pp

prq

pu

where ;16716

,42

1 2224

2 tqrqprp

xpqr

qp, and r are arbitrary

constants.

Fig. 4.1. Solution 1u explains the soliton solution corresponding to the parameter values

.2,1,1 rqp Solitons are particular kinds of solitary waves. An amazing chattel of

solitons is that it maintains its individuality upon interacting with other solitons.

,cot2

csc2

2

3

cot2

csc2

2

32

8

222222

2

pp

prq

pu

,22sin22cos

2sin12sec4

2

3

22sin22cos

2sin12sec4

2

32

8

2222

3

pp

prq

pu

,22cos22sin

2cos12csc4

2

3

22cos22sin

2cos12csc4

2

32

8

2222

4

pp

prq

pu

107

,2cos2sin

csc2

2

3

2cos2sin

csc2

2

32

8

2222

5

pp

prq

pu

,2

8

22cos22sin2sin22cos

2sin2sin2cos4

2

3

22cos22sin2sin22cos

2sin2sin2cos4

2

3

2

2

222222

222

222222

222

6

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu

,2

8

22cos22sin2sin22cos

2sin2sin2cos4

2

3

22cos22sin2sin22cos

2sin2sin2cos4

2

3

2

2

222222

222

222222

222

7

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu

where RBA 0, are constants and gratify the condition 022 BA .

,4cossin4cos22

sin2cossec2

2

3

4cossin4cos22

sin2cossec2

2

32

8

2

222

2

222

22

8

prqp

p

prqp

pprq

pu

,cossin4cos22

cos2sincsc2

2

3

cossin4cos22

cos2sincsc2

2

32

8

2

222

2

222

22

9

pprqp

p

pprqp

pprq

pu

,2cos22sin122cos2

22sin22cos2sin12sec2

2

3

2cos22sin122cos2


2

32

8

2

22

2

22

22

10

prqp

p

prqp

pprq

pu

,22sin22sin122cos2

22cos22sin2csc2

2

3

22sin22sin122cos2

22cos22sin2csc2

2

32

8

2

2

2

2

22

11

rqppprq

p

rqppprq

pprq

pu

108

.cossin4cos22

cos2sincsc2

2

3

cossin4cos22

cos2sincsc2

2

32

8

2

222

2

222

22

12

pprqp

p

pprqp

pprq

pu

Family 2. When 042 qrp and ),0or (0 qrqp the soliton-like solutions are:

,tanh2

sec2

2

3

tanh2

sec2

2

32

8

222222

13

p

h

p

hprq

pu

where ;16716

,42

1 2224

2 tqrqprp

xqrp

qp, and r are arbitrary

constants.

,coth2

csc2

2

3

coth2

csc2

2

32

8

222222

14

p

h

p

hprq

pu

,22sinh22cosh

2sinh12sec4

2

3

22sinh22cosh

2sinh12sec4

2

32

8

2222

15

ip

ih

ip

ihprq

pu

,22cosh22sinh

2cosh12csc4

2

3

22cosh22sinh

2cosh12csc4

2

32

8

2222

16

p

h

p

hprq

pu

,2/coth2/tanh12/cosh2

2/sec

2

3

2/coth2/tanh12/cosh2

2/sec

2

32

8

2

2

22

2

222

17

p

h

p

hprq

pu

,

2cosh222sinh2sin

2cosh2sinh4

2

3

2cosh222sinh2sin

2cosh2sinh4

2

32

8

2

22

222

22

2222

18

ABApBpABA

BABAA

ABApBpABA

BABAAprq

pu

,

2cosh222sinh2sin

2cosh2sinh4

2

3

2cosh222sinh2sin

2cosh2sinh4

2

32

8

2

22

222

22

2222

19

ABApBpABA

BABAA

ABApBpABA

BABAAprq

pu

109

where RBA 0, are constants and gratify the condition .022 BA

,coshsinh2

sec2

2

3

coshsinh2

sec2

2

32

8

2222

20

p

h

p

hprq

pu

Fig. 4.2. Solution 20u explains the soliton solution for corresponding to the values

.2,1,3 rqp

,sinhcosh2

csc2

2

3

sinhcosh2

csc2

2

32

8

2222

21

p

h

p

hprq

pu

,22sinh22cosh

2sinh12sec4

2

3

22sinh22cosh

2sinh12sec4

2

32

8

2222

22

ip

ih

ip

ihprq

pu

,22cosh22sinh

2cosh12csc4

2

3

22cosh22sinh

2cosh12csc4

2

32

8

2222

23

p

h

p

hprq

pu

.sinhcosh2

2csc2

2

3

sinhcosh2

2csc2

2

32

8

2222

24

p

h

p

hprq

pu

Family 3. When 0r and ,0qp the solutions are

,sinhcosh

sinhcosh

2

3

sinhcosh

sinhcosh

2

32

8

22

25

pppd

ppp

pppd

pppprq

pu

,

sinhcosh2

3

sinhcosh2

32

8

22

26

pppd

dp

pppd

dpprq

pu


110

Fig. 4.3. Illustrates the form of the kink-type solution of 26u for .0,2,2 rqp Kink

waves are traveling waves which take place from one asymptotic state to another. The

Family 4. When 0q and ,0 pr the solution is

,2

3

2

32

8

2

11

2

27

dq

q

dq

qprq

pu


Fig. 4.4. Explains the shape of the soliton solution 27u corresponding to the parameter

values .0,3,0 rqp

111

Example 4.2.2. The Sixth-Order Boussinesq Equation

Consider the sixth-order Boussinesq equation [198]

.09045)(153015 62

222

234 xxxxxxxxxtt uuuuuuuuuuuu (4.10)

The wave variable (1.30) converts (4.10) in an ordinary differential equation as follows:

,09045513015 622242 uuuuuuuuuuuuV (4.11)

where 4u and 6u denotes the fourth order and sixth order derivatives of u with respect

to . Integrating (4.11) once with respect to yields:

,01515451 )5(22 uuuuuuVuC (4.12)

where C is a constant of integration. Now, considering the homogeneous balance

between the highest order derivative and nonlinear term in (4.12), we obtain .2n

According to step 3 of section 1.4.4, the trial solution of (4.12) becomes:

,)(2

210

G

Ga

G

Gaau (4.13)

where ia are constants to be indomitable and G is the solution of the generalized Riccati

equation (4.5). Using (4.5), equation (4.13) can be rewritten as:

.)()()( 212


Proceeding as before, the following solution set is attained:

.0,1168,2,2,3

4

3

1 222421

20 CrqrqppVaparqpa (4.15)

Substituting (4.15) into (4.13), the following solution is obtained:

,223

4

3

1)(

2

2

G

G

G

Gprqpu (4.16)

112

where .1168 2224 trqqrppx

Now on the basis of the solutions of (4.5), the families of exact solutions are:


,tan2

sec22

tan2

sec22

3

4

3

12

22222

1

ppprqpu

where ,1168,42

1 22242 trqqrppxpqr and rqp ,, are arbitrary

constants.

,cot2

csc22

cot2

csc22

3

4

3

12

22222

2

ppprqpu

,22sin22cos

2sin12sec42

22sin22cos

2sin12sec42

3

4

3

12

222

3

ppprqpu

,22cos22sin

2cos12csc42

22cos22sin

2cos12csc42

3

4

3

12

222

4

ppprqpu

,2cos2sin

csc22

2cos2sin

csc22

3

4

3

1222

25

ppprqpu

,

3

4

3

1

22cos22sin2sin22cos

2sin2sin2cos42

22cos22sin2sin22cos

2sin2sin2cos42

2

2

222222

222

222222

222

6

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu

,

3

4

3

1

22cos22sin2sin22cos

2sin2sin2cos42

22cos22sin2sin22cos

2sin2sin2cos42

2

2

222222

222

222222

222

7

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu


113

,4cossin4cos22

sin2cossec22

4cossin4cos22

sin2cossec22

3

4

3

1

2

222

2

222

22

8

prqp

p

prqp

pprqpu

,cossin4cos22

cos2sincsc22

cossin4cos22

cos2sincsc22

3

4

3

1

2

222

2

222

22

9

pprqp

p

pprqp

pprqpu

,2cos22sin122cos2


2cos22sin122cos2


3

4

3

1

2

22

2

22

22

10

prqp

p

prqp

pprqpu

,22sin22sin122cos2

22cos22sin2csc22

22sin22sin122cos2

22cos22sin2csc22

3

4

3

1

2

2

2

2

22

11

rqppprq

p

rqppprq

pprqpu

.cossin4cos22

cos2sincsc22

cossin4cos22

cos2sincsc22

3

4

3

1

2

222

2

222

22

12

pprqp

p

pprqp

pprqpu

Family 2. When 042 qrp and ),0or (0 qrqp the soliton-like solutions are:

,tanh2

sec22

tanh2

sec22

3

4

3

12

22222

13

p

h

p

hprqpu

where ,1168,42

1 22242 trqqrppxqrp and rqp ,, are arbitrary

constants.

114

Fig. 4.5. Solution 13u elucidates the soliton for .1,2,3 rqp The particular types

0)(),(),( uuu as , .tVx Solitons have a amazing property that

,coth2

csc22

coth2

csc22

3

4

3

12

22222

14

p

h

p

hprqpu

,22sinh22cosh

2sinh12sec42

22sinh22cosh

2sinh12sec42

3

4

3

1222

215

ip

ih

ip

ihprqpu

,22cosh22sinh

2cosh12csc42

22cosh22sinh

2cosh12csc42

3

4

3

12

222

16

p

h

p

hprqpu


2/sec2


2/sec2

3

4

3

1

2

2

22

2

222

17

p

h

p

hprqpu

,

2cosh222sinh2sin

2cosh2sinh42

2cosh222sinh2sin

2cosh2sinh42

3

4

3

1

2

22

222

22

2222

18

ABApBpABA

BABAA

ABApBpABA

BABAAprqpu

115

,

2cosh222sinh2sin

2cosh2sinh42

2cosh222sinh2sin

2cosh2sinh42

3

4

3

1

2

22

222

22

2222

19

ABApBpABA

BABAA

ABApBpABA

BABAAprqpu


,coshsinh2

sec22

coshsinh2

sec22

3

4

3

1222

220

p

h

p

hprqpu

,sinhcosh2

csc22

sinhcosh2

csc22

3

4

3

1222

221

p

h

p

hprqpu

,22sinh22cosh

2sinh12sec42

22sinh22cosh

2sinh12sec42

3

4

3

1222

222

ip

ih

ip

ihprqpu

,22cosh22sinh

2cosh12csc42

22cosh22sinh

2cosh12csc42

3

4

3

1222

223

p

h

p

hprqpu

.sinhcosh2

2csc22

sinhcosh2

2csc22

3

4

3

12

222

24

p

h

p

hprqpu

Family 3. When 0r and ,0qp the solutions are:

,sinhcosh

sinhcosh2

sinhcosh

sinhcosh2

3

4

3

12

225

pppd

ppp

pppd

pppprqpu

,

sinhcosh2

sinhcosh2

3

4

3

12

226

pppd

dp

pppd

dpprqpu


Family 4. When 0q and ,0 pr the solution is:

,223

4

3

12

11

227

dq

q

dq

qprqpu


116

Fig. 4.6. Demonstrates the singular soliton solution 27u for .0,1,0 rqp

Example 4.2.3. The Fifth Order CDGSK Equation

Consider the following fifth order CDGSK equation [43] in the form


The fifth order CDGSK equation (4.17) is a completely integrable. The first four

conserved densities of equation (4.17) are:

.4

3

4

9

4

1

,3

1

,

,

22

244

233

2

1

xx

x

uuuuT

uuT

T

uT

(4.17a)

Now, using the traveling wave variable tVxutxu ),(),( that converts (4.17)

into an ODE as follows:

,05 25 uuuuuuuuV (4.18)

where 5u indicates the ordinary derivative of u with respect to . Integrating (4.18)

once with respect to yields:

117

,03

55 34 uuuuuVC (4.19)

where C is a constant of integration. Considering the homogeneous balance between

4u and 3u in (4.19), we deduced that .2n

According to step 3 of section 1.4.4, the trial solution of (4.19) takes the form

.)(2

210

G

Ga

G

Gaau (4.20)


.)()()( 212


Proceeding as before, the solution of the obtained system is

.3

2

3

128832

,168,6,6,4

6334222

222421

20

prqqrprqpC

rqrqppVaparqpa

(4.22)

Substituting (4.22) into (4.20), the following solution is obtained:

,/6/64)(22 GGGGprqpu (4.23)

where .168 2224 trqrqppx

Now on the basis of the solutions of (4.5), the families of exact solutions are:

Family 1. When 042 qrp and ),0or (0 qrqp the periodic form solutions are

,tan2

sec26

tan2

sec264

22222

21

ppprqpu

where ;168,42

1 22242 trqqrppxpqr rqp ,, are arbitrary constants.

118

Fig. 4.7. Solution 1u explains the soliton for .1,1,1 rqp The particular types of

solitary waves are solitons. The soliton solution is a spatially confined solution, hence, 0)(),(),( uuu as , .tVx An amazing chattel of solitons is that it

maintains its individuality upon interacting with other solitons.

,cot2

csc26

cot2

csc264

22222

22

ppprqpu

,22sin22cos

2sin12sec46

22sin22cos

2sin12sec464

222

23

ppprqpu

,22cos22sin

2cos12csc46

22cos22sin

2cos12csc464

222

24

ppprqpu

,2cos2sin

csc26

2cos2sin

csc264

2222

5

ppprqpu

,4

22cos22sin2sin22cos

2sin2sin2cos46

22cos22sin2sin22cos

2sin2sin2cos46

2

2

222222

222

222222

222

6

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu

119

,4

22cos22sin2sin22cos

2sin2sin2cos46

22cos22sin2sin22cos

2sin2sin2cos46

2

2

222222

222

222222

222

7

rqp

BApBApAABBAA

BAABBAA

BApBApAABBAA

BAABBAApu


,4cossin4cos22

sin2cossec26

4cossin4cos22

sin2cossec264

2

222

2

222

22

8

prqp

p

prqp

pprqpu

,cossin4cos22

cos2sincsc26

cossin4cos22

cos2sincsc264

2

222

2

222

22

9

pprqp

p

pprqp

pprqpu

,2cos22sin122cos2


2cos22sin122cos2


2

22

2

22

22

10

prqp

p

prqp

pprqpu

,22sin22sin122cos2

22cos22sin2csc26

22sin22sin122cos2

22cos22sin2csc264

2

2

2

2

22

11

rqppprq

p

rqppprq

pprqpu

,cossin4cos22

cos2sincsc26

cossin4cos22

cos2sincsc264

2

222

2

222

22

12

pprqp

p

pprqp

pprqpu


,tanh2

sec2

2

3

tanh2

sec264

22222

213

p

h

p

hprqpu

120

where ,168,42

1 22242 trqqrppxqrp qp, and r are arbitrary

constants.

Fig. 4.8. Solution 13u elucidates the soliton for .5.0,1,2 rqp The particular types of

solitary waves are solitons. The soliton solution is a spatially confined solution, hence, 0)(),(),( uuu as , .tVx Solitons have an amazing property that

it keeps its characteristics upon interacting with other solitons.

,coth2

csc26

coth2

csc264

22222

214

p

h

p

hprqpu

Fig. 4.9. Illustrates the singular soliton solution of 14u for .1,2,3 rqp

121

,22sinh22cosh

2sinh12sec46

22sinh22cosh

2sinh12sec464

2222

15

ip

ih

ip

ihprqpu

,22cosh22sinh

2cosh12csc46

22cosh22sinh

2cosh12csc464

2222

16

p

h

p

hprqpu


2/sec6


2/sec64

2

2

22

2

222

17

p

h

p

hprqpu

,

2cosh222sinh2sin

2cosh2sinh46

2cosh222sinh2sin

2cosh2sinh464

2

22

222

22

2222

18

ABApBpABA

BABAA

ABApBpABA

BABAAprqpu

,

2cosh222sinh2sin

2cosh2sinh46

2cosh222sinh2sin

2cosh2sinh464

2

22

222

22

2222

19

ABApBpABA

BABAA

ABApBpABA

BABAAprqpu


,coshsinh2

sec26

coshsinh2

sec264

222

220

p

h

p

hprqpu

,sinhcosh2

csc26

sinhcosh2

csc264

2222

21

p

h

p

hprqpu

,22sinh22cosh

2sinh12sec46

22sinh22cosh

2sinh12sec464

2222

22

ip

ih

ip

ihprqpu

,22cosh22sinh

2cosh12csc46

22cosh22sinh

2cosh12csc464

222

223

p

h

p

hprqpu

,sinhcosh2

2csc26

sinhcosh2

2csc264

222

224

p

h

p

hprqpu

122

Family 3. When 0r and ,0qp the solutions are

,sinhcosh

sinhcosh6

sinhcosh

sinhcosh64

2

225

pppd

ppp

pppd

pppprqpu

,

sinhcosh6

sinhcosh64

2

226

pppd

dp

pppd

dpprqpu


Family 4. When 0q and ,0 pr the solution is

,664

2

11

227

dq

q

dq

qprqpu


Fig. 4.10. Shows the shape of the soliton solution 27u for .0,2,0 rqp

Example 4.2.4. The (3 + 1)-Dimensional Modified KdV-ZK Equation

Consider the (3+1)-dimensional modified KdV-ZK equation [232] in the form

,02 zzxyyxxxxxt uuuuuu (4.24)

where is a nonzero constant parameter.

Now, using the wave transformation tVzyxutzyxu ),(),,,( that converts

(4.24) in an ODE as follows

123

,032 uuuuV (4.25)

Integrating (4.25) once with respect yields

,033

1 3 uuVuC (4.26)

where C is an integral constant which is to be determined later. Now, cconsidering the

homogeneous balance between u and 3u in (4.26), we deduced that .1n

According to step 3 of section 1.4.4, the trial solution of (4.26) takes the form:

./)( 10 GGaau (4.27)


).()( 110 GqGrpaau (4.28)

Proceeding in a similar manner as before yields:

,2

36,12

2

3,

2

6,

2

3 210

pqriCrqpV

iip (4.29)


,tan2

sec2

2

6

2

3 22

1

p

iipu

Fig.4.11. Solution 1u represents the exact periodic traveling wave solution for

.9,2,1,1 rqp The traveling wave solutions which are periodic in nature like

)cos( tx are called periodic solutions.

124

where ;122

3,4

2

1 22 trqpzyxpqr

qp, and r are arbitrary constants.

,cot2

csc2

2

6

2

3 22

2

p

iipu

,22sin22cos

2sin12sec4

2

6

2

3 2

3

p

iipu

,22cos22sin

2cos12csc4

2

6

2

3 2

4

p

iipu

,2cos2sin

csc2

2

6

2

3 2

5

p

iipu

,

2

3

22cos22sin2sin22cos

2sin2sin2cos4

2

6

222222

222

6

ip

BApBApAABBAA

BAABBAAiu

,

2

3

22cos22sin2sin22cos

2sin2sin2cos4

2

6

222222

222

7

ip

BApBApAABBAA

BAABBAAiu


,4cossin4cos22

sin2cossec2

2

6

2

3222

2

8

prqp

piipu

,cossin4cos22

cos2sincsc2

2

6

2

3222

2

9

pprqp

piipu

,2cos22sin122cos2


2

6

2

322

2

10

prqp

piipu

,22sin22sin122cos2

22cos22sin2csc2

2

6

2

32

2

11

rqppprq

piipu

,cossin4cos22

cos2sincsc2

2

6

2

3222

2

12

pprqp

piipu

125

Family 2. When 042 qrp and ),0or (0 qrqp the solutions are

,tanh2

sec2

2

6

2

3 22

13

p

hiipu

where ;122

3,4

2

1 22 trqpzyxqrp

qp, and r are arbitrary constants.

Fig. 4.12. Solution 13u explains the soliton for .4,1,3,4 rqp The particular

types of solitary waves are solitons. The soliton solution is a spatially confined solution, hence 0)(),(),( uuu as , .tVx An amazing chattel of solitons is

that it maintains its individuality upon interacting with other solitons.

,coth2

csc2

2

6

2

3 22

14

p

hiipu

,22sinh22cosh

2sinh12sec4

2

6

2

3 2

15

ip

ihiipu

,22cosh22sinh

2cosh12csc4

2

6

2

3 2

16

p

hiipu


2/sec

2

6

2

32

22

17

p

hiipu

126

,

2cosh222sinh2sin

2cosh2sinh4

2

6

2

322

222

18

ABApBpABA

BABAAiipu

,

2cosh222sinh2sin

2cosh2sinh4

2

6

2

322

222

19

ABApBpABA

BABAAiipu


,coshsinh2

sec2

2

6

2

3 2

20

p

hiipu

,sinhcosh2

csc2

2

6

2

3 2

21

p

hiipu

,22sinh22cosh

2sinh12sec4

2

6

2

3 2

22

ip

ihiipu

,22cosh22sinh

2cosh12csc4

2

6

2

3 2

23

p

hiipu

,sinhcosh2

2csc2

2

6

2

3 2

24

p

hiipu

Family 3. When 0r and ,0qp the solutions are:

,sinhcosh

sinhcosh

2

6

2

325

pppd

pppiipu

,

sinhcosh2

6

2

326

pppd

dpiipu


Family 4. When 0q and ,0 pr the solution is:

,2

6

2

3

1

27

dq

qiipu


127

Discussion

In this chapter, by using the alternative )/( GG -expansion method, copious traveling

wave solutions of diverse nonlinear evolution equations have been attained in a consistent

way, in which the generalized Riccati equation is used as an auxiliary equation. The

attained solutions are noteworthy for the elucidation of some realistic physical

phenomena. It is revealed that the alternative )/( GG -expansion method together with

the generalized Riccati equation offers a sophisticated mathematical tool for solving

nonlinear partial differential equations. The graphical depictions together with the

numerical results divulge the complete fidelity and lofty effectiveness of the algorithm.

128

Chapter 5

Traveling Wave Solutions of Nonlinear Partial Differential

Equations by )/1,/( GGG -Expansion Method

129

5.1. Introduction

It has recently become more important to obtain exact solutions of nonlinear partial

differential equations through symbolic computer algebra that facilitates complex and

dreary algebraic computations. Calculating the exact numerical solutions, particularly,

traveling wave solutions of nonlinear differential equations in mathematics and

mathematical physics plays an important role in soliton theory [45, 222]. Mathematical

modeling of physical systems is normally explained with nonlinear differential equations.

These equations are mathematical models of complex physical phenomena that arise in

engineering and mechanics. Various effective methods have been developed to recognize

the mechanism of these physical models, which help the engineers and physicists to a

great extent by helping them to ensure that they have the adequate knowledge to solve the

physical problems. In the past decades, there has been significant progress in the

development of constructive methods for obtaining exact solutions of NLEEs. These

methods are; Modified Simple Equation [107, 264], First Integration [206] and Variation

of Parameter [171, 172] methods.

In 2010, Li et al. [116] developed a relatively new technique )1,/( GGG -expansion

method to find the traveling wave solutions of the Zakharov equations. It is worth

mentioning that recently Zayed and Abdelaziz [267] applied )1,/( GGG -expansion

method for finding traveling wave solutions of the nonlinear (3 + 1)-dimensional

Kadomtsev-Petviashvili equation. Zayed et al. [261] also applied this technique to solve

nonlinear KdV-mKdV equation. The basic idea of )1,/( GGG -expansion method is that

traveling wave solutions of nonlinear evolution equations can be expressed by a

polynomial in two variables )/( GG and )/1( G in which )(GG satisfies a second

order linear ordinary differential equation as an auxiliary equation,

,)()( GG

and sets GG /)( and ,/1)( G then

,2 .

This chapter comprises the applications of )1,/( GGG -expansion method for traveling

wave solutions of nonlinear partial differential equations [30, 188, 200]. The proposed

130

technique provides useful and new traveling wave solutions of the nonlinear PDEs, which

differ from the solutions, obtained by using the other existing techniques.


In this section some examples are given to elaborate the effectiveness of the well known

direct computational method called )/1,/( GGG -expansion method. The graphical

representations together with the numerical results disclose the complete trustworthiness

and high efficiency of the method.

Example 5.2.1. The Positive Gardner-KP Equation

Consider the positive Gardner-KP equation [200] as follows,

.066 2 yyxxxxxxt uuuuuuu (5.1)

Bin and Qiang [38] derived the conservation laws for the positive Gardner-KP equation

(5.1) by using the adjoint equation and symmetries.

Using the wave transformation tVyxPP ),( into (5.1) and integrating once

and setting the constant of integration to zero yields:

.0661 2 uuuuuuV (5.2)

According to step 3 of section 1.4.5, the solution of (5.2) is can be articulated by a power

series in and as follows

M

n

nn

M

n

nn bau

1

1

0

, (5.3)

where Mnan ,,3,2,1 and Mnbn ,,3,2,1 are constants to be determine later

and , are given by,

,)(

G

G

,

1)(

G (5.4)

where the following equation

,0 GG (5.5)

131

is used as an auxiliary equation.

The general solutions of (5.5) are followed by Li [116].

Applying the balancing principle betweenu and uu 2 in (5.2), we deduced that 1M .

Therefore, the trail solution becomes,

,110 baau (5.6)

where ,0a 1a and 1 b are constants which are to be determined later. Here three cases are

to be discussed as follows:

Case 1. When 0 (Hyperbolic function solutions).

If ,0 inserting (5.6) along with (1.40) and (1.41) into (5.2), the left-hand side of (5.2)

becomes a polynomial in and . Setting each coefficient of this polynomial to zero

yields a system of algebraic equations in ,0a ,,, , 11 ba V and as follows:

,06618

: 13122

2114

aa

ba

,0612

121261212: 2

1210222

231

22

210

22

1

22

121

22

2103

aaa

bbabbaba

,068

6612612324

:

11101

311

20222

22211

22

11

22

110

22

21

22

22112

Vaaaaa

aaabababaaaba

,0612

12612612: 2

1210222

331

22

221

22

21

21

22

21

22

22101

aaa

bbbabba

,06186

: 112122

313

bbab

(5.7)

,066651212

24126122424:

,0126122442

:

1110211

2011

21

210

222

2231

22

21

22

231

22

21

22

210

22

21

21

1131111022

2112

Vbbbaababbaaa

bbbbbaba

baaabaaba

,066512

6241261824

:

1110111110

120222

32211

22

211

22

31

22

2211

22

21101

Vaaaabaabaa

aababaababaa

132

.06

261261263

:

1110

211

20222

23211

22

211

22

2110

22

3211

22

2210

Vaaaa

aaabababaabaa

Solving the above system of equations (5.7), following two solution sets are obtained:

Set 1.

.,)(2

)2

,)(

,0,12

1

22

22

2223

22

1122

0

BAV

baa

(5.8)

Now, the traveling wave solution of (5.1) becomes:

,coshsinh

)(1

2

1,,

22

221

BAtyxu

where .)(2

)222

2223

tyx

In particular, if setting 0 and 0,0 BA in 1u , the solitary wave solution is

Fig. 5.1. Solution 11u explains the soliton solution for .,1 xy Solitons are particular

kinds of solitary waves. An amazing chattel of solitons is that it maintains its individuality upon interacting with other solitons.

133

and if setting 0 and 0,0 BA in 1u , the following solitary wave solution is,

.csc2

1,,2

1 htyxu

Fig. 5.2. Solution 21u explains the soliton solution for .,1 xy

Set 2.

.,2

1,

4,

2

1,

2

1 2222

110 BAVbiaa

(5.9)


,coshsinh4

coshsinh

sinhcosh

2

1

2

1,,

22

2/3

2

BA

BA

BAityxu

where .2

1tyx

In particular, by setting 0 and 0,0 BA in 2u , the following solitary solution is

obtained,

,sec2

1tanh

2

1

2

1,,1

2 hityxu

134

Fig. 5.3. Solution 12u explains the soliton solution for .,1 xy

while, if setting 0 and 0,0 BA in 2u , the following solitary solution is,

.csccoth2

1

2

1,,2

2 htyxu

Case 2. When 0 (Trigonometric function solutions).

If ,0 substituting (5.6) along with (1.40) and (1.42) into (5.2), the left- hand side of

(5.2) becomes a polynomial in . and Setting each coefficient of this polynomial to

zero yields a system of algebraic equations in ,0a ,,, , 11 ba V and as follows:

,06618

: 13122

2114

aa

ba

,0612

12661212: 2

1210222

231

22

21

22

1

22

210

22

1213

aaa

bbbbaba

,068

6612123246

:

11101

311

20222

22211

22

110

22

21

22

2211

22

112

Vaaaaa

aaababaaababa

,0612

12661212:

21

210

222

331

22

221

22

21

22

2210

22

21

211

aaa

bbbbaba

,06186

: 112122

313

bbab

(5.10)

135

,066651212

24126122424:

,0126122442

:

1110211

2011

21

210

222

2231

22

21

22

231

22

21

22

21

21

22

210

1131111022

2112

Vbbbaababbaaa

bbbbbaba

baaabaaba

,066512

6241218246

:

1110111110

120222

32211

22

211

22

2211

22

2110

22

311

Vaaaabaabaa

aababababaaa

.06

261266312

:

1110

211

20222

23211

22

211

22

3211

22

221

22

21100

Vaaaa

aaabababaabaa

Solving the above system of algebraic equations, following two solution sets are

obtained:

Set 3.

.,2

)2

,,0,12

1

22

22

2223

22

1122

0

BAV

baa

(5.11)

Now, the traveling wave solution of (5.1) becomes

,cossin

12

1,,

22

223

BAtyxu

where .

2

222

2223

tyx

In particular, if setting 0 and 0,0 BA in 3u , the solitary wave solution is,

,csc2

1,,1

3 ityxu

while, if setting 0 and 0,0 BA in 3u , the following solitary wave solution is,

.sec2

1,,2

3 ityxu

136

Fig. 5.4. Solution 23u shows the soliton solution for .,2 xy Solitons are particular

kinds of solitary waves. Solitons have a notable property that it keeps its uniqueness upon

Set 4.

.,2

1,

4,

2

1,

2

1 2222

110 BAVbiaa

(5.12)

,cossin4

cossin

sincos

2

1

2

1,,

22

2/3

4

BA

BA

BAityxu

where .2

1tyx

In particular, by setting 0 and 0,0 BA in 4u , the solitary wave solution is,

,sectan2

1

2

1,,1

4 ityxu

while, if setting 0 and 0,0 BA in 4u , the solitary wave solution is,

.csccot2

1

2

1,,2

4 ityxu

137

Fig. 5.5. Solution 24u shows the soliton solution for .,2 xy Solitons are particular

kinds of solitary waves. Solitons have a notable property that it keeps its uniqueness upon interacting with other solitons.

Case 3. When 0 (Rational function solutions).

If ,0 inserting (5.6) along with (1.40) and (1.43) into (5.2), the left-hand side of (5.2)

becomes a polynomial in . and Setting each coefficient of the resulted polynomial to

zero yields a system of algebraic equations in ,0a , , 11 ba andV as follows:

,0662

18: 1

312

2114

aa

BA

ba

,0612

2

12

2

12

2

12

2

6

2

6: 2

121022

31

2

121

2

210

2

21

2

13

aaaBA

b

BA

ba

BA

ba

BA

b

BA

b

,066

2

12

2

6

2

3

2

12: 11101

2022

2211

2

11

2

21

2

1102

VaaaaaaBA

ba

BA

ba

BA

a

BA

baa

,012612242

42:

,06182

6:

113111102

2112

11212

313

baaabaaBA

ba

bbaBA

b

(5.13)

,066612

2

24

2

24

2

12

2

12

2

24:

1110211

20

210

22

231

2

21

21

2

21

2

21

2

210

Vbbbaabaaa

BA

b

BA

ba

BA

b

BA

b

BA

ba

138

.06

62

24

2

12

2

24

2

6:

1110

12022

3211

2

211

2

2110

2

311

Vaaaa

aaBA

ba

BA

ba

BA

baa

BA

a

Solving the system (5.13), the following two solution sets are obtained:

Set 5.

.

22

32,2,0,

21

2

12

222

112

0BA

BAVBAba

BAa

(5.14)


,2

12

21

2

1,,

2

2

205

BABA

BAatyxu

where .

22

322

22

tBA

BAyx

Set 6.

.2

1,2

2

1,

2

1,

2

1 2110 VBAbiaa (5.15)

,2

12

2

1

22

1

2

1,,

2

2

26

BABA

BA

Aityxu

where .2

1tyx

Example 5.2.2. The (2 + 1)-Dimensional CBS Equation

Now, consider the (2 + 1)-dimensional CBS equation [188] in the form

.024 zxxzxxzxxxtx uuuuuu (5.16)

This equation is used to describe the interaction of a Riemann wave promulgating along

the z-axis with a long wave along the x-axis.

Using the traveling wave transformation ,),(),,( tVwzkxutzxu that converts

(5.16) into an ordinary differential equation

139

.06 2)4(3 uuwkuwkukV (5.17)

Integrating (5.17) with respect to and setting the constant of integration to zero yields:

,0)(3 223 uwkuwkukV (5.18)

where primes denote the derivatives with respect to . Applying the homogeneous

balancing principle between u and 2)(u , we deduced that 1M . According to step 3 of

section 1.4.5, the solution of (5.18) can be expressed by a finite power series in and

as follows:

,110 baau (5.19)

where ,0a 1a and 1 b are constants to be determined later. The following cases are to be

discussed as follows:

Case 1. Hyperbolic function solutions .0

If ,0 inserting (5.19) along with (1.40) and (1.41) into (5.18), the left-hand side of

(5.18) becomes a polynomial in . and Setting each coefficient of this polynomial to

zero yields a system of algebraic equations in ,0a ,,, , 11 ba V and as follows:

,0333

86:

,066

:

,03

63:

22

221

2

22

21

3

22

221

2

132

12

12

221

3

2211

23

22

21

2

132

124

wbkwakwakwakawkVka

wbkbwak

wbkwakwak

,066

:22

21

3

22

211

21

wbkbwak (5.20)

,06

612:

,066:

22

21

221

21

32

13

1123

wbkawkwak

wbkbwak

140

.066

56:

,033

23:

22

31

3

22

321

2

132

12

10

22

221

3

22

2221

22

1322

12

11

wakwakwakawkVka

wakwakwakwakVka

Solving the above system of algebraic equations, the following solution set is obtained:

.,,,, 22222

1100 BAwkVkbkaaa

(5.21)


,coshsinh

coshsinh

sinhcosh,,

222/3

01

BA

k

BA

BAkatzxu

where .2 twkwzkx

In particular, by taking 0 and 0,0 BA in 1u , the solution is,

,sectanh,, 011 hikatzxu

Fig. 5.6. Shows the form of the exact kink-type solution of 11u for ,0,1 0

.0,1 zk Kink waves are traveling waves which take place from one asymptotic state

to another. The kink solution comes close to a constant at infinity.

and if taking 0 and 0,0 BA in 1u , the solitary wave solution is,

.csccoth,, 021 hkatzxu

141

Fig. 5.7. Shows the form of the singular kink solution of 21u for .0,1,0,1 0 zk

Case 2. If ,0 substituting (5.19) along with (1.40) and (1.42) into (5.18), the equation

(5.18) becomes a polynomial and setting each coefficient to zero yields a system of

equations in unknowns and after solving, the following solution is obtained:

.,,,, 22222

1100 BAwkVkbkaaa

(5.22)


,cossin

cossin

sincos,,

222/3

02

BA

k

BA

BAkatzxu

where .2 twkwzkx

In particular, if setting 0 and 0,0 BA in 2u , the solitary solution is,

,sectan,, 012 katzxu

142


,0,1 0 .1,,1 wxzk The traveling wave solutions which are periodic in

nature like )cos( tx are called periodic solutions.

Similarly, taking 0 and 0,0 BA in 2u , the solitary wave solution is,

.csccot,, 022 katzxu

Example 5.2.3. The Modified Benjamin-Bona-Mahony Equation

Now, consider the nonlinear MBBM equation [30] in the form

,02 txxxxt uuuuu (5.23)

which was first derived to depict an ballpark figure for surface long waves in nonlinear

dispersive media. This equation can also characterize the hydro magnetic waves in cold

plasma and acoustic-gravity waves in compressible fluids [154].

The wave variable (1.35) converts (5.23) into an ODE as follows

,01 2 uVuuuV (5.24)

Applying the homogeneous balancing principle between u and ,2uu we deduced

that 1M . According to step 3 of section 1.4.5, the solution of (5.24) can be expressed

by a finite power series in and as

,110 baau (5.25)

where ,0a 1a and 1 b are constants to be determined later. Now, the three cases are as

follows:

143

Case 1. Hyperbolic function solutions .0

If ,0 substituting (5.25) along with (1.40) and (1.41) into (5.24), the left-hand side of

(5.24) becomes a polynomial and setting each coefficient to zero yields a system of

equations in ,0a ,,, , 11 ba V and as follows:

,0226

122:

,03

6:

121

210122222

2312

103

22

211

131

4

babaVbb

aa

baVaa

,0262

122:

,08

2234

1:

21

21

21

221022222

2312

101

11120

311

222

21

221

1102

1122

122

2

baVbbab

aa

VaVaaaaa

abbaaVaab

,07

412:

,036:

22

211

110131

2

22

31

1211

3

babaaVaa

bbaVb

(5.26)

02312

12:

,04

5221

44121

:

21

2110

2

22222

23211

120

1

222

2231

1121

2101

201

12

1023

12

121

2122

abbaVba

aaV

bbVbaaababV

bbabbaVb

.04

251

6431

:

222

32211

110120

31

2110

221122

0

babaaaaVV

Vabaaba

Solving the above system, we obtain the following two solution sets:

144

Set 1.

.,2)1()2(

3,

2)1()2(

)(2

),(2)1()2(

12,0

22

22022

22

22

3211

BAaV

ba

(5.27)

Now the traveling wave solution of (5.23) becomes:

,coshsinh

)(2)1()2(

12

2)1()2(

3, 22

32221

BA

txu

where .2)1()2(

)(222

22

tx

In particular, if taking 0 and 0,0 BA in 1u , the solitary wave solution is,

,sec1

6,1

1

htxu

Fig. 5.9. Solution 11u explains the soliton for .1 The particular types of solitary

waves are solitons. The soliton solution is a spatially confined solution, hence 0)(),(),( uuu as , .tVx An amazing chattel of solitons is that it

maintains its individuality upon interacting with other solitons.

145

while, if setting 0 and 0,0 BA in 1u , the solitary wave solution is,

.csc1

6,2

1

htxu

Set 2.

.,2

2,

)2(

)(3,

2

3,0 22

22

110 BAVbaa

(5.28)

Now in this result the traveling wave solution of (5.23) becomes:

,coshsinh

)2(

)(3

coshsinh

sinhcosh

2

3,

222/3

2

BA

BA

BAtxu

where .2

2tx

In particular, by setting 0 and 0,0 BA in 2u , the solution is,

,sec2

3tanh

2

3,1

2

htxu

Fig. 5.10. Shows the form of the exact kink-type solution of 12u for .1 Kink waves

are traveling waves which take place from one asymptotic state to another. The kink solution comes close to a constant at infinity.

146

while, if setting 0 and 0,0 BA in 2u , the solitary solution is,

.csccoth2

3,2

2

htxu

Case 2. If ,0 substituting (5.25) along with (1.40) and (1.42) into (5.24), the left-

hand side of (5.24) becomes a polynomial and setting each coefficient to zero yields a

system of equations in unknowns and after solving the system, two solutions are

obtained:

Set 3.

.,2)1()2(

)(2,

2)1()2(

3

),(2)1()2(

12,0

22

22

22

220

22

3211

BAVa

ba

(5.29)


,cossin

)(2)1()2(

12

2)1()2(

3, 22

32223

BA

txu

where .2)1()2(

)(222

22

tx

In particular, by taking 0 and 0,0 BA in 3u , the solitary solution is,

,csc1

6,1

3

txu

147

Fig. 5.11. Solution 13u explains the soliton for .1 Solitons are particular kinds of

solitary waves. An amazing chattel of solitons is that it maintains its individuality upon interacting with other solitons.

while, if setting 0 and 0,0 BA in 3u , the solitary solution is,

.sec1

6,2

3

txu

Fig.5.12. Solution 23u corresponds to the periodic traveling wave solution for .2

Set 4.

.,2

2,

)2(

)(3,

2

3,0 22

22

110 BAVbaa

(5.30)

Now, in this result the traveling wave solution of (5.23) becomes:

148

,cossin

)2(

)(3

cossin

sincos

2

3,

222/3

4

BA

BA

BAtxu

where .2

2tx

In particular, by setting 0 and 0,0 BA in 4u , the solitary solution is,

,sectan2

3,1

4

txu

Fig.5.13. Solution 14u corresponds to the periodic traveling wave solution for .1

while, if setting 0 and 0,0 BA in 4u , the solution is,

.csccot2

3,2

4

txu

Case 3. When ,0 by analogous computations, the following two solution sets are

obtained:

Set 5.

.342

)2(2

),2(342

12,0,

342

3

22

2

2

2211220

BA

BAV

BABA

baBA

a

(5.31)

149


,2

42

342

3,

2

2

225

BA

BA

BAtxu

where .342

)2(222

2

tBA

BAx

Set 6.

.1,2

)2(3,

2

3,0

2

110

VBA

baa

(5.32)

Now, the travelling wave solution of (5.23) becomes:

,2

2

2

3,

2

2

6

BA

BAAtxu

where .tx

Discussion

In this chapter )1,/( GGG -expansion method is applied to obtain new solitary wave

solutions of nonlinear evolution equations. The crucial advantage of )/1,/( GGG -

expansion method aligned with the fundamental )/( GG -expansion method is that it

offers abundant, new and general exact traveling wave solutions. These exact solutions

are of great significance in enlightening the internal mechanism of the complicated

physical phenomena. The closed-form of solutions assists the numerical researchers in

comparing the exactness of their outcomes and also helps them in the stability analysis.

150

Chapter 6

Applications of Novel )/( GG -Expansion Method to

Nonlinear Evolution Equations

151

6.1. Introduction

Currently, scientists believe that nonlinear science is the most important medium for the

fundamental understanding of nature. Many complex physical phenomena are often

described and modeled by nonlinear evolution equations. Therefore, the exact solutions

of the nonlinear evolution equations have become increasingly essential. These are not

only regarded as valuable tool in examining the accuracy of computational dynamics, but

also facilitate the researchers to readily understand the requisites of complex physical

phenomenon, for example, collision of two solitary solutions. The analytical solutions of

nonlinear evolutions are of crucial importance to help and to understand the internal

mechanism of the physical phenomena. In mathematics and physics, a soliton is a self-

reinforcing solitary wave, a wave packet that maintains its profile, while traveling at a

constant speed. In the past years, many powerful and direct methods have been developed

to discover special solutions, such as; Variational Iteration [173], Exp-Function [179],

Differential Transform [236] methods and others [235, 250].

Quite a lot of researchers carried out further research in order to ascertain the efficiency

and accuracy of )/( GG -expansion method and to enlarge the assortment of its

applicability. Alam et al. [26] introduced a novel )/( GG -expansion method for finding

the more general solutions of Boussinesq equation.

In this chapter the soliton solutions of nonlinear evolution equations [199, 216, 219, 231,

232] are obtained by applying the novel )/( GG -expansion method. This method offers

all types of solutions including; trigonometric function, hyperbolic function and rational

solutions. To illustrate the innovation, steadiness of the projected method, numbers of

nonlinear evolution equations are solved and abundant new families of exact solutions are

obtained.


In this section, a promising and powerful technique, called novel )/( GG -expansion

method is used to solve the equations [199, 216, 219, 231, 232], which arise in

mathematical physics, engineering sciences and applied mathematics. The accuracy of

the method verifies the efficiency and reliability of novel )/( GG -expansion method.

152

Example 6.2.1. The ZK-BBM Equation

Consider the ZK-BBM equation [199] in the form

.02 txxxxt buauuuu (6.1)

The ZK-BBM belongs to the integrable systems. Adem and Khalique [20] derived the

new conservation laws for the ZK-BBM equation by using Ibragimov conservation

theorem and the multiplier method.

Using the traveling wave variable (1.45) that converts (6.1) into an ODE and upon

integrating yields:

,01 12 CubVuauV (6.2)

where 1C is a constant of integration.

2m . Then according to step 2 of section 1.4.6, the trial solution of (6.2) becomes

.)()()()()(2

210

1

1

2

2

kkkku (6.3)

Using (6.3) in (6.2), the left hand side transforms into polynomials in iGGk )/(

and iGGk

)/( .,,2,1,0 Ni Equating all coefficients with the same power of

the resulted polynomials to zero, a set of algebraic equations (which are omitted here for

the sake of simplicity) for 0 , 1 , 2 , 1 , 2 , k , 1C andV is obtained. Solving the

indomitable set of algebraic equations, following three solution sets are obtained:

Set 1.

,

162

2a

CbV

,

2426 2

1a

kAACCkkCbV (6.4)

,0,0,,),

1882412121212(2

1

212

22220

kkVVbVAV

bVBbVBCbVCkbVkAbVkbVkACkbVCa

,)1232

881616(4

1

2222

42222222222222221

VVCBVb

AVbBCAVbBAVbBVbCBVba

C

where k, V, A, B and C are arbitrary constants.

153

Set 2.

,18824121212122

1 222220 bVAVbVBbVBCbVCkbVkACbVkAbVkkbVC

a

,24332226 3222332

1 BCKCkACkkAAkkBAkCBka

bV

,2222226 24322342242

2 BCkABkCkACkkAAkkBBkkCa

bV

,VV ,kk ,02 ,01 (6.5)

,)1232

881616(4

1

2222

42222222222222221

VVCBVb

AVbBCAVbBAVbBVbCBVba

C


Set 3.

,

162

2a

CbV ,1882

2

1 20 VbVBCbVBbVA

a

,81632816)1(8

3 4222222

22 ABABCBBCACBCa

,VV ,)1(2

C

Ak ,01 ,01 (6.6)

,)12128128

51216256256(4

1

2222222

22242222222221

VVBAVbBCAVb

CBVbAVbBVbCBVba

C

where V, A, B and C are arbitrary constants.

Inserting equations (6.4)- (6.6) in (6.3), the following solutions of (6.1) are obtained:

.)/(16

)/(

2426)18

82412121212(2

1)(

22

22

22221

GGka

CbVGGk

a

kAACCkkCbVbVAVbVB

bVBCbVCkbVkAbVkbVkACkbVCa

u

(6.7)

154

.)/()22

2222(6

)/()2433

222(6

)2418

812121212(2

1)(

22

4322342242

13222

33222

2222

GGkBCkABk

CkACkkAAkkBBkkCa

bV

GGkBCkABCkACkkAAk

kBkkCa

bVbVCkbVAVbVB

bVBCbVkAbVkbVkACkbVCa

u

(6.8)

,)/()1(2

81632816)1(8

3

)/()1(2

161882

2

1)(

2

4222222

2

222

3

GGC

A

ABABCBBCACBCa

GGC

A

a

CbVVbVBCbVBbVA

au

(6.9)

where Vtx ; V, A, B and C are arbitrary constants.

Substituting the solutions )(G of (1.49) in (6.7) and simplifying, the following solutions

are obtained:

When 0442 BBCA and 0)1( CA (or 0)1( CB ), the solutions are

,)2/tanh(12

116

)2/tanh(12

1

2426)18

82412121212(2

1)(

22

22

222211

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAVbVB


u

155

Fig. 6.1. Shows the graph of cuspon of 11u for .1,1,3,1,2,1,1 VbakCBA

Unlike Peakons, where the derivative at the climax fluctuates only by a sign, the derivative at the hop of cuspons deviates.

,)2/coth(12

116

)2/coth(12

1

2426)18

82412121212(2

1)(

22

22

222221

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAVbVB


u

Fig. 6.2. Exact singular soliton solution 21u for ,2,1,1 CBA .1,1,1,1 Vbak

156

,)(sec)tanh(12

116

)(sec)tanh(12

1

2426)18

82412121212(2

1)(

22

22

222231

hiAC

ka

CbV

hiAC

k

a

kAACCkkCbVbVAVbVB


u

,)(csc)coth(12

116

)(csc)coth(12

1

2426)18

82412121212(2

1)(

22

22

222241

hAC

ka

CbV

hAC

k

a

kAACCkkCbVbVAVbVB


u

,)4/coth()4/tanh(214

116

)4/coth()4/tanh(214

1

2426)18

82412121212(2

1)(

22

22

222251

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAVbVB


u

,)sinh(

)cosh()(

12

116

)sinh(

)cosh()(

12

1

2426)18

82412121212(2

1)(

2222

22

22

222261

BF

FHFA

Ck

a

CbV

BF

FHFA

Ck

a

kAACCkkCbVbVAVbVB


u

157

,)sinh(

)cosh()(

12

116

)sinh(

)cosh()(

12

1

2426)18

82412121212(2

1)(

2222

22

22

222271

BF

FHFA

Ck

a

CbV

BF

FHFA

Ck

a

kAACCkkCbVbVAVbVB


u

where F and H are real constants.

,

)2/cosh()2/sinh(

)2/cosh(216

)2/cosh()2/sinh(

)2/cosh(2

2426)18

82412121212(2

1)(

22

22

222281

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAVbVB


u

,

)2/sinh()2/cosh(

)2/sinh(216

)2/sinh()2/cosh(

)2/sinh(2

2426)1

882412121212(2

1)(

22

22

222291

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

158

Fig. 6.3. Solution 91u explains the soliton for ,2,1,1 CBA .1,1,3,1 Vbak

The particular types of solitary waves are solitons. The soliton solution is a spatially confined solution, hence 0)(),(),( uuu as , .tVx An amazing

chattel of solitons is that it maintains its individuality upon interacting with other solitons

,

)cosh()sinh(

)cosh(216

)cosh()sinh(

)cosh(2

2426)1

882412121212(2

1)(

22

22

2222101

iA

Bk

a

CbV

iA

Bk

a

kAACCkkCbVbVAV


u

,

)sinh()cosh(

)sinh(216

)sinh()cosh(

)sinh(2

2426)1

882412121212(2

1)(

22

22

2222111

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

159


,)2/tan(12

116

)2/tan(12

1

2426)1

882412121212(2

1)(

22

22

2222121

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAV


u


,2,1,1 CBA .1,1,3,1 Vbak Periodic solutions are traveling wave

solutions that are periodic such as ).cos( tx

,)2/cot(12

116

)2/cot(12

1

2426)1

882412121212(2

1)(

22

22

2222131

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAV


u

160

Fig.6.5. Solution 131u represents the exact singular periodic traveling wave solutions for

2,1,1 CBA .1,1,3,2 Vbak

,)sec()tan(12

116

)sec()tan(12

1

2426)1

882412121212(2

1)(

22

22

2222141

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAV


u

,)(csc)cot(12

116

)(csc)cot(12

1

2426)1

882412121212(2

1)(

22

22

2222151

hAC

ka

CbV

hAC

k

a

kAACCkkCbVbVAV


u

161

,)4/cot()4/tan(214

116

)4/cot()4/tan(214

1

2426)1

882412121212(2

1)(

22

22

2222161

AC

ka

CbV

AC

k

a

kAACCkkCbVbVAV


u

,)sin(

)cos()(

12

116

)sin(

)cos()(

12

1

2426)1

882412121212(2

1)(

2222

22

22

2222171

BF

FHFA

Ck

a

CbV

BF

FHFA

Ck

a

kAACCkkCbVbVAV


u

,)sin(

)cos()(

12

116

)sin(

)cos()(

12

1

2426)1

882412121212(2

1)(

2222

22

22

2222181

BF

FHFA

Ck

a

CbV

BF

FHFA

Ck

a

kAACCkkCbVbVAV


u

where F and H are real constants such that .022 HF

162

,

)2/cos()2/sin(

)2/cos(216

)2/cos()2/sin(

)2/cos(2

2426)1

882412121212(2

1)(

22

22

2222191

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

,

)2/sin()2/cos(

)2/sin(216

)2/sin()2/cos(

)2/sin(2

2426)1

882412121212(2

1)(

22

22

2222201

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

,

)cos()sin(

)cos(216

)cos()sin(

)cos(2

2426)1

882412121212(2

1)(

22

22

2222211

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

.

)2/sin()2/cos(

)2/sin(216

)2/sin()2/cos(

)2/sin(2

2426)1

882412121212(2

1)(

22

22

2222221

A

Bk

a

CbV

A

Bk

a

kAACCkkCbVbVAV


u

163

When 0B and 0)1( CA , the solution is

,)sinh()cosh()1(

16

)sinh()cosh()1(

2426)1

882412121212(2

1)(

2

1

1

2

1

1

22

2222231

AAcC

Ack

a

CbV

AAcC

Ack

a

kAACCkkCbVbVAV


u

,)sinh()cosh()1(

))sinh()(cosh(16

)sinh()cosh()1(

))sinh()(cosh(

2426)1

882412121212(2

1)(

2

1

2

1

22

2222241

AAcC

AAAk

a

CbV

AAcC

AAAk

a

kAACCkkCbVbVAV


u

where 1c is an arbitrary constant.

When 0 BA and 0)1( C , the solution of (6.1) is

,

)1(

116

)1(

12426)1

882412121212(2

1)(

2

2

2

2

22

2222251

cCk

a

CbV

cCk

a

kAACCkkCbVbVAV


u


Inserting the solutions )(G of (1.49) in (6.8) and simplifying, the following solutions

are obtained

164


,)2/tanh(

12

1)2

22222(6

)2/tanh(12

1)243

3222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32

2233222

22212

AC

kBCk

ABkCkACkkAAkkBBkkCa

bV

AC

kBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV

bVBbVBCbVkAbVkbVkACkbVCa

u

where Vtx ; k, A, B and C are arbitrary constants.

,)2/coth(

12

1)2

22222(6

)2/coth(12

1)243

3222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32

2233222

22222

AC

kBCk

ABkCkACkkAAkkBBkkCa

bV

AC

kBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV


u

.))(sec)(tanh(

12

1)2

22222(6

))(sec)(tanh(12

1)243

3222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32

2233222

22232

hiAC

kBCk

ABkCkACkkAAkkBBkkCa

bV

hiAC

kBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV


u

165


,)2/tan(

12

1)2

22222(6

)2/tan(12

1)243

3222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32

2233222

222122

AC

kBCk

ABkCkACkkAAkkBBkkCa

bV

AC

kBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV


u

,)2/cot(

12

1)2

22222(6

)2/cot(12

1)243

3222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32

2233222

222132

AC

kBCk

ABkCkACkkAAkkBBkkCa

bV

AC

kBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV


u

.))sec()(tan(

12

1)2

22222(6

))sec()(tan(12

1)2

433222(6

)24

18812121212(2

1)(

2

2

4322342242

1

32223322

2222142

AC

kBCk

ABkCkACkkAAkkBBkkCa

bV

AC

kBCk

ABCkACkkAAkkBkkCa

bVbVCk

bVAVbVBbVBCbVkAbVkbVkACkbVCa

u

166


,)1(

1)2

22222

(6

)1(

1)243

3222(6

)24

18812121212(2

1)(

2

2

2

43223422

42

1

3

32

2233222

222252

cCkBCk

ABkCkACkkAAkkBBk

kCa

bV

cCkBCkABCkACk

kAAkkBkkCa

bVbVCkbVAV


u


Finally, substituting the solutions )(G of (1.49) in (6.9) and simplifying, the following

solutions are obtained:


,))2/tanh(()1(2

1)81632

816()1(8

3))2/tanh((

)1(2

1

161882

2

1)(

2

4222

222

2

2

221

3

CABABCB

BCACBCaC

a

CbVVbVBCbVBbVA

au

where Vtx ; A, B and C are arbitrary constants.

,))2/coth(()1(2

1)81632

816()1(8

3))2/coth((

)1(2

1

161882

2

1)(

2

4222

222

2

2

222

3

CABABCB

BCACBCaC

a

CbVVbVBCbVBbVA

au

167

.)(sec)(tanh()1(2

1)81632

816()1(8

3)(sec)(tanh(

)1(2

1

161882

2

1)(

2

4222

222

2

2

223

3

hiC

ABABCB

BCACBCa

hiC

a

CbVVbVBCbVBbVA

au

For simplicity others families of exact solutions are omitted.

When 0442 BBCA and 0)1( CA (or 0)1( CB ), the solutions are,

,))2/tan(()1(2

1)81632

816()1(8

3))2/tan((

)1(2

1

161882

2

1)(

2

4222

222

2

2

2

2123

CABABCB

BCACBCaC

a

CbVVbVBCbVBbVA

au

Fig.6.6. Solution 123u represents the exact singular periodic traveling wave solution for

168

,))2/cot(()1(2

1)81632

816()1(8

3))2/cot((

)1(2

1

161882

2

1)(

2

4222

222

2

2

2

2133

CABABCB

BCACBCaC

a

CbVVbVBCbVBbVA

au

.)(sec)(tan()1(2

1)81632

816()1(8

3)(sec)(tan(

)1(2

1

161882

2

1)(

2

4222

222

2

2

2

2143

CABABCB

BCACBCaC

a

CbVVbVBCbVBbVA

au

When 0)1( C and 0 BA , the solution of (6.1) is

,)1(

1

)1(2)81632

816()1(8

3

)1(

1

)1(2

161882

2

1)(

2

2

4222

222

2

2

4

2225

3

cCC

AABABCB

BCACBCacCC

A

a

CbVVbVBCbVBbVA

au


Example 6.2.2. The Symmetric Regularized Long Wave Equation

Consider the following SRLW equation [231] in the form

,0 xxtttxxtxxtt uuuuuuu (6.10)

which occurs in numerous physical applications including ion sound waves in plasma.

This equation is symmetrical with respect to x and t. It occurs in many nonlinear

problems of mathematical physics and applied mathematics.

Now, using the traveling wave variable ),(uu Vtx in (6.10) and integrating

twice yields:

,02

11 1

222 CuVuVuV (6.11)

169

where 1C is an integration constant. Now considering the homogeneous balance between

u and 2u in (6.11), we obtain .2m Then according to step 2 of section 1.4.6, the trial

solution of (6.11) becomes

.)()()()()(2

210

1

1

2

2

kkkku (6.12)

Proceeding as before, the following three solution sets are obtained:

Set 1.

,1122

2 CV ,24212 21 kAACCkkCV ,02 ,01

,1

1882412121212 222220

VABBCCkkAkkACkCV ,kk ,VV

,2

1132881616

22422222

3

1V

VCBABCABABCBV

C (6.13)


Set 2.

,1

1882412121212 222220

VABBCCkkACkAkkCV ,02 ,01

,243322212 32223321 BCkCkACkkAAkkBAkCBkV ,VV

,22222212 243223422422 BCkABkCkACkkAAkkBBkkCV

,2

1132881616

22422222

3

1V

VCBABCABABCBV

C ,kk (6.14)


Set 3.

,162

2 CV ,1

1828 20

VBABCV ,VV ,

)1(2

C

Ak ,01

,81632816)1(4

3 4222222

22 ABABCBBCACBC

,01 (6.15)


Substituting equations (6.13)-(6.15) in (6.12), the following solutions are attained:

170

.)/(112

)/(24212

11882412121212)(

22

2

222221

GGkCV

GGkkAACCkkCV

VABBCCkkAkkACkCVu

(6.16)

.)/()222

222(12)/(

243322212

11882412121212)(

224

3223422421

3222332

222222

GGkBCkABkCk

ACkkAAkkBBkkCVGGk

BCkCkACkkAAkkBAkCBkV

VABBCCkkACkAkkCVu

(6.17)

,)/()1(2

81632816)1(4

3

)/()1(2

161

)1828()(

2

4222222

2

2

223

GGC

A

ABABCBBCACBC

GGC

ACV

VBABCVu

(6.18) where Vtx ; V, A, B and C are arbitrary constants.

Inserting the solutions )(G of (1.49) in (6.16) and simplifying, we get


,)2/tanh(12

1112

)2/tanh(12

124212

11882412121212)(

2

2

2

2222211

AC

kCV

AC

kkAACCkkCV

VABBCCkkAkkACkCVu

171

Fig. 6.7. Solution 11u shows the graph of cuspon for .1,1,2,1,1 VkCBA

,)2/coth(12

1112

)2/coth(12

124212

11882412121212)(

2

2

2

2222221

AC

kCV

AC

kkAACCkkCV

VABBCCkkAkkACkCVu

Fig. 6.8. Illustrates exact singular soliton solution 21u for .1,1,2,1,1 VkCBA

172

,)(sec)tanh(12

1112

)(sec)tanh(12

124212

11882412121212)(

2

2

2

2222231

hiAC

kCV

hiAC

kkAACCkkCV

VABBCCkkAkkACkCVu

,)(csc)coth(12

1112

)(csc)coth(12

124212

11882412121212)(

2

2

2

2222241

hAC

kCV

hAC

kkAACCkkCV

VABBCCkkAkkACkCVu


1112

)4/coth()4/tanh(214

1

24212

11882412121212)(

2

2

2

2222251

AC

kCV

AC

k

kAACCkkCV

VABBCCkkAkkACkCVu

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

1

24212

11882412121212)(

222

2

22

2

2222261

BF

FHFA

CkCV

BF

FHFA

Ck

kAACCkkCV

VABBCCkkAkkACkCVu

173

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

1

24212

11882412121212)(

222

2

22

2

2222271

BF

FHFA

CkCV

BF

FHFA

Ck

kAACCkkCV

VABBCCkkAkkACkCVu


,)2/cosh()2/sinh(

)2/cosh(2112

)2/cosh()2/sinh(

)2/cosh(224212

11882412121212)(

2

2

2

2222281

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

,)2/sinh()2/cosh(

)2/sinh(2112

)2/sinh()2/cosh(

)2/sinh(224212

11882412121212)(

2

2

2

2222291

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

174

Fig. 6.9. Solution 91u elucidates the soliton for ,2,1,1 CBA .1,1 Vk The

particular types of solitary waves are solitons. The soliton solution is a spatially confined solution, hence 0)(),(),( uuu as , .tVx An amazing chattel of

solitons is that it maintains its individuality upon interacting with other solitons.

,)cosh()sinh(

)cosh(2112

)cosh()sinh(

)cosh(224212

11882412121212)(

2

2

2

22222101

iA

BkCV

iA

BkkAACCkkCV

VABBCCkkAkkACkCVu

.)sinh()cosh(

)sinh(2112

)sinh()cosh(

)sinh(224212

11882412121212)(

2

2

2

22222111

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

175

When 0442 BBCA and 0)1( CA (or 0)1( CB ), the solution are

,)2/tan(12

1112

)2/tan(12

124212

11882412121212)(

2

2

2

22222121

AC

kCV

AC

kkAACCkkCV

VABBCCkkAkkACkCVu


,2,1,1 CBA .1,1 Vk The traveling wave solutions which are periodic in nature

like )cos( tx are called periodic solutions.

,)2/cot(12

1112

)2/cot(12

124212

11882412121212)(

2

2

2

22222131

AC

kCV

AC

kkAACCkkCV

VABBCCkkAkkACkCVu

176

,)sec()tan(12

1112

)sec()tan(12

1

24212

11882412121212)(

2

2

2

22222141

AC

kCV

AC

k

kAACCkkCV

VABBCCkkAkkACkCVu

,)(csc)cot(12

1112

)(csc)cot(12

1

24212

11882412121212)(

2

2

2

22222151

hAC

kCV

hAC

k

kAACCkkCV

VABBCCkkAkkACkCVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

11882412121212)(

222

2

22

2

22222161

BF

FHFA

CkCV

BF

FHFA

Ck

kAACCkkCV

VABBCCkkAkkACkCVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

11882412121212)(

222

2

22

2

22222171

BF

FHFA

CkCV

BF

FHFA

Ck

kAACCkkCV

VABBCCkkAkkACkCVu

177

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

11882412121212)(

222

2

22

2

22222181

BF

FHFA

CkCV

BF

FHFA

Ck

kAACCkkCV

VABBCCkkAkkACkCVu


,)2/cos()2/sin(

)2/cos(2112

)2/cos()2/sin(

)2/cos(224212

11882412121212)(

2

2

2

22222191

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

,)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(224212

11882412121212)(

2

2

2

22222201

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

,)cos()sin(

)cos(2112

)cos()sin(

)cos(224212

11882412121212)(

2

2

2

22222211

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

178

.)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(224212

11882412121212)(

2

2

2

22222221

A

BkCV

A

BkkAACCkkCV

VABBCCkkAkkACkCVu

When 0B and 0)1( CA , the solutions are

,)sinh()cosh()1(

112

)sinh()cosh()1(24212

11882412121212)(

2

1

12

1

12

22222231

AAcC

AckCV

AAcC

AckkAACCkkCV

VABBCCkkAkkACkCVu

Fig.6.11. 231u represents the bell-shaped 2sec h solitary traveling wave solution for

,0,1 BA 1,1,2 VkC separated by infinite wings or infinite tails.

179

,)sinh()cosh()1(

))sinh()(cosh(112

)sinh()cosh()1(

))sinh()(cosh(24212

11882412121212)(

2

1

2

1

2

22222241

AAcC

AAAkCV

AAcC

AAAkkAACCkkCV

VABBCCkkAkkACkCVu


When 0 BA and 0)1( C , the solution is

,)1(

1112

)1(

124212

11882412121212)(

2

2

2

2

2

22222251

cCkCV

cCkkAACCkkCV

VABBCCkkAkkACkCVu


Similarly, inserting the solutions )(G of (1.49) in (6.17) and simplifying, we get

different families of exact solutions of (6.10) as follows.


,)2/tanh(

12

1

22222212

)2/tanh(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

2222212

AC

k

BCkABkCkACkkAAkkBBkkCV

AC

k


VABBCCkkACkAkkCVu


180

,)2/coth(

12

1

22222212

)2/coth(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

2222222

AC

k


AC

k


VABBCCkkACkAkkCVu

.))(sec)(tanh(

12

1

22222212

))(sec)(tanh(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

2222232

hiAC

k


hiAC

k


VABBCCkkACkAkkCVu

The other exact solutions of (6.10) are omitted for convenience.


,)2/tan(

12

1

22222212

)2/tan(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

22222122

AC

k


AC

k


VABBCCkkACkAkkCVu

181


,2,1,1 CBA .1,2 Vk The traveling wave solutions which are periodic in

nature like )cos( tx are called periodic solutions.

,)2/cot(

12

1

22222212

)2/cot(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

22222132

AC

k


AC

k


VABBCCkkACkAkkCVu

.))sec()(tan(

12

1

22222212

))sec()(tan(12

1

243322212

11882412121212)(

2

24322342242

1

3222332

22222142

AC

k


AC

k


VABBCCkkACkAkkCVu

182

When 0 BA and 0)1( C , the solution of (6.10) is,

,)1(

1)2222

22(12)1(

1

243322212

11882412121212)(

2

2

243

22342242

1

2

3222332

22222252

cCkBCkABkCkACk

kAAkkBBkkCVcC

k


VABBCCkkACkAkkCVu


The other exact solutions of (6.10) are omitted here for the sake of simplicity.

Lastly, surrogating the solutions )(G of (1.49) in (6.18) and simplifying, the solutions

of (6.10) are:


,))2/tanh(()1(2

1

81632816)1(4

3

))2/tanh(()1(2

116

1)1828()(

2

4222222

2

2

2213

C

ABABCBBCACBC

CCV

VBABCVu


,))2/coth(()1(2

1

81632816)1(4

3

))2/coth(()1(2

116

1)1828()(

2

4222222

2

2

2223

C

ABABCBBCACBC

CCV

VBABCVu

183

.)(sec)(tanh()1(2

1

81632816)1(4

3

)(sec)(tanh()1(2

116

1)1828()(

2

4222222

2

2

2233

hiC

ABABCBBCACBC

hiC

CVV

BABCVu

The others exact solutions are omitted for the sake of simplicity.


,))2/tan(()1(2

1

81632816)1(4

3

))2/tan(()1(2

116

1)1828()(

2

4222222

2

2

22123

C

ABABCBBCACBC

CCV

VBABCVu

Fig.6.13. 123u represents the exact singular periodic traveling wave solution for

.1,2,1,1 VCBA

184

,))2/cot(()1(2

1

81632816)1(4

3

))2/cot(()1(2

116

1)1828()(

2

4222222

2

2

22133

C

ABABCBBCACBC

CCV

VBABCVu

.)(sec)(tan()1(2

1

81632816)1(4

3

)(sec)(tan()1(2

116

1)1828()(

2

4222222

2

2

22143

C

ABABCBBCACBC

CCV

VBABCVu


,)1(

1

)1(281632816

)1(4

3

)1(

1

)1(216

1)1828()(

2

2

4222222

2

2

2

22253

cCC

AABABCBBCACB

CcCC

ACV

VBABCVu


Other exact solutions of (6.10) are omitted for convenience.

Example 6.2.3. The Boussinesq System

Consider the well-known Boussinesq system [216] in the form,

,0

,02

xxxxt

xt

buuav

vu (6.19)

This system of equations is used to model two-way propagation of certain water waves in

a homogeneous horizontal channel filled with an irrotational and inviscid liquid [184].

Now, using the traveling wave variable (1.45) that converts (6.19) into a system of ODEs

.0,,0

,0

2

baubuavV

vuV (6.20)

185

Integrating the system (6.20) once, we find

,

,

12 CubauVv

Vuv

(6.21)

where 1C is an integration constant. Substituting first equation of the system (6.21) into

second equation, we obtain

.0122 CubauuV (6.22)

Now, considering the homogeneous balance between u and 2u in (6.22), we obtain

.2m Therefore, the trial solution becomes

2

210

1

1

2

2 )()()()()(

kkkku .(6.23)

Proceeding as before, three solution sets are obtained as follows:

Set 1.

,

162

2a

Cb

,

2426 2

1a

kAACCkkCb ,VV ,kk ,01 ,02

,2

882412121212 222222

0a

VbAbBbBCbCkbkAbkbkACkbC

,4

32881616 42242222222222

1a

VCBbAbBCAbBAbBbCBbC

(6.24)


Set 2.

,2

882412121212 222222

0a

VbAbBbBCbCkbkACbkAbkkbC

,24332226 3222332

1a

BCkCkACkkAAkkBAkCBkb

,

2222226 24322342242

2a

BCkABkCkACkkAAkkBBkkCb

,02 01 , ,VV ,kk (6.25)

,4

32881616 42242222222222

1a

VCBbAbBCAbBAbBbCBbC


186

Set 3.

,

162

2a

Cb ,

2

828 22

0a

VbBbAbBC ,VV ,

)1(2

C

Ak ,01

,

)1(8

8163281632

4222222

2

Ca

ABABCBBCACBb ,01 (6.26)

,4

12812851216256256 42222224222222

1a

VBAbBCAbCBbAbBbCBbC


Substituting equations (6.24)-(6.26) in (6.23) yields

.)/(16

)/(2426

2

882412121212)(

222

222222

1

GGka

CbGGk

a

kAACCkkCb

a

VbAbBbBCbCkbkAbkbkACkbCu

(6.27)

.)/()222

222(6

)/(

24332226

2

882412121212)(

224

3223422421

3222332

222222

2

GGkBCkABkCk

ACkkAAkkBBkkCa

bGGk

a


a

VbAbBbBCbCkbkACbkAbkkbCu

(6.28)

,)/(

)1(2)1(8

816328163

)/()1(2

16

2

828)(

2

2

4222222

2222

3

GGC

A

Ca

ABABCBBCACBb

GGC

A

a

Cb

a

VbBbAbBCu

(6.29)

where Vtx ; k, V, A, B and C are arbitrary constants.

187

Inserting the solutions )(G of (1.49) in (6.27) and simplifying, the solutions are


,)2/tanh(12

116

)2/tanh(12

12426

2

882412121212)(

22

2

22222211

AC

ka

Cb

AC

ka

kAACCkkCb

a


Fig. 6.14. 11u shows the graph of cuspon for .1,1,1,1,2,1,1 VbakCBA

Cuspons are diverse forms of solitons where solution shows cusps at their crests. Unlike

peakons where the derivatives at the peak differ only by a sign, the derivatives at the

jump of a cuspon diverge.

,)2/coth(12

116

)2/coth(12

12426

2

882412121212)(

22

2

22222221

AC

ka

Cb

AC

ka

kAACCkkCb

a


188

,)(sec)tanh(12

116

)(sec)tanh(12

12426

2

882412121212)(

22

2

22222231

hiAC

ka

Cb

hiAC

ka

kAACCkkCb

a


,)(csc)coth(12

116

)(csc)coth(12

12426

2

882412121212)(

22

2

22222241

hAC

ka

Cb

hAC

ka

kAACCkkCb

a



116

)4/coth()4/tanh(214

12426

2

882412121212)(

22

2

22222251

AC

ka

Cb

AC

ka

kAACCkkCb

a


,)sinh(

)cosh()(

12

116

)sinh(

)cosh()(

12

1

2426

2

882412121212)(

2222

22

2

22222261

BF

FHFA

Ck

a

Cb

BF

FHFA

Ck

a

kAACCkkCb

a


189

,)sinh(

)cosh()(

12

116

)sinh(

)cosh()(

12

1

2426

2

882412121212)(

2222

22

2

22222271

BF

FHFA

Ck

a

Cb

BF

FHFA

Ck

a

kAACCkkCb

a



,

)2/cosh()2/sinh(

)2/cosh(216

)2/cosh()2/sinh(

)2/cosh(22426

2

882412121212)(

22

2

22222281

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


,

)2/sinh()2/cosh(

)2/sinh(216

)2/sinh()2/cosh(

)2/sinh(22426

2

882412121212)(

22

2

22222291

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


190

Fig.6.15. Solution 91u explains the soliton for ,2,1,1 CBA .1,3,1,1 Vbak

Solitons are meticulous kinds of solitary waves. The soliton solution is a spatially confined solution, hence 0)(),(),( uuu as , .tVx An amazing

chattel of solitons is that it maintains its individuality upon interacting with other solitons.

,

)cosh()sinh(

)cosh(216

)cosh()sinh(

)cosh(22426

2

882412121212)(

22

2

222222101

iA

Bk

a

Cb

iA

Bk

a

kAACCkkCb

a


.

)sinh()cosh(

)sinh(216

)sinh()cosh(

)sinh(22426

2

882412121212)(

22

2

222222111

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


191


,)2/tan(12

116

)2/tan(12

12426

2

882412121212)(

22

2

222222121

AC

ka

Cb

AC

ka

kAACCkkCb

a


Fig.6.16. 121u represents the exact periodic traveling wave solution for

,1,2,2,1,1 akCBA .1,3 Vb

,)2/cot(12

116

)2/cot(12

12426

2

882412121212)(

22

2

222222131

AC

ka

Cb

AC

ka

kAACCkkCb

a


192

,)sec()tan(12

116

)sec()tan(12

1

2426

2

882412121212)(

22

2

222222141

AC

ka

Cb

AC

k

a

kAACCkkCb

a


,)(csc)cot(12

116

)(csc)cot(12

1

2426

2

882412121212)(

22

2

222222151

hAC

ka

Cb

hAC

k

a

kAACCkkCb

a


,)sin(

)cos()(

12

116

)sin(

)cos()(

12

1

2426

2

882412121212)(

2222

22

2

222222161

BF

FHFA

Ck

a

Cb

BF

FHFA

Ck

a

kAACCkkCb

a


,)sin(

)cos()(

12

116

)sin(

)cos()(

12

1

2426

2

882412121212)(

2222

22

2

222222171

BF

FHFA

Ck

a

Cb

BF

FHFA

Ck

a

kAACCkkCb

a


193

,)sin(

)cos()(

12

116

)sin(

)cos()(

12

1

2426

2

882412121212)(

2222

22

2

222222181

BF

FHFA

Ck

a

Cb

BF

FHFA

Ck

a

kAACCkkCb

a



,

)2/cos()2/sin(

)2/cos(216

)2/cos()2/sin(

)2/cos(22426

2

882412121212)(

22

2

222222191

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


,

)2/sin()2/cos(

)2/sin(216

)2/sin()2/cos(

)2/sin(22426

2

882412121212)(

22

2

222222201

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


,

)cos()sin(

)cos(216

)cos()sin(

)cos(22426

2

882412121212)(

22

2

222222211

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a


194

.

)2/sin()2/cos(

)2/sin(216

)2/sin()2/cos(

)2/sin(22426

2

882412121212)(

22

2

222222221

A

Bk

a

Cb

A

Bk

a

kAACCkkCb

a



,)sinh()cosh()1(

16

)sinh()cosh()1(

2426

2

882412121212)(

2

1

1

2

1

12

222222231

AAcC

Ack

a

Cb

AAcC

Ack

a

kAACCkkCb

a


Fig.6.17. 231u represents the bell-shaped 2sec h solitary traveling wave solution for

1,3,1,1,2,0,1 VbakCBA alienated by infinite wings or infinite tails.

195

,)sinh()cosh()1(

))sinh()(cosh(16

)sinh()cosh()1(

))sinh()(cosh(2426

2

882412121212)(

2

1

2

1

2

222222241

AAcC

AAAk

a

Cb

AAcC

AAAk

a

kAACCkkCb

a




,

)1(

116

)1(

12426

2

882412121212)(

2

2

2

2

2

222222251

cCk

a

Cb

cCk

a

kAACCkkCb

a



Surrogating the solutions )(G of (1.49) in (6.27) and simplifying, the solutions are

When 0442 BBCA and 0)1( CA (or 0)1( CB )

,)2/tanh(

12

1

2222226

)2/tanh(12

1

24332226

2

882412121212)(

2

24322342242

1

3222332

22222212

AC

k

a


AC

k

a


a



196

,)2/coth(

12

1

2222226

)2/coth(12

1

24332226

2

882412121212)(

2

24322342242

1

3222332

22222222

AC

k

a


AC

k

a


a


.))(sec)(tanh(

12

1

2222226

))(sec)(tanh(12

1

24332226

2

882412121212)(

2

24322342242

1

3222332

22222232

hiAC

k

a


hiAC

k

a


a


The other solutions of (6.19) are omitted for convenience.

When 0442 BBCA and 0)1( CA (or 0)1( CB ),

,)2/tan(

12

1)2222

22(6

)2/tan(12

1

24332226

2

882412121212)(

2

24322

342242

1

3222332

222222122

AC

kBCkABkCkACkkA

AkkBBkkCa

bA

Ck

a


a


197

,)2/cot(

12

1

2222226

)2/cot(12

1

24332226

2

882412121212)(

2

24322342242

1

3222332

222222132

AC

k

a


AC

k

a


a



,2,1,1 CBA .1,5,1,1 Vbak The traveling wave solutions which are

periodic in nature like )cos( tx are called periodic solutions.

.))sec()(tan(

12

1

2222226

))sec()(tan(12

1

24332226

2

882412121212)(

2

24322342242

1

3222332

222222142

AC

k

a


AC

k

a


a


198


,)1(

1)222

222(6

)1(

1

24332226

2

882412121212)(

2

2

24

322342242

1

2

3222332

222222252

cCkBCkABkCk

ACkkAAkkBBkkCa

b

cCk

a


a



Finally, substituting the solutions )(G of (1.49) in (6.27) and simplifying, the solutions

of (6.19) are:


,))2/tanh(()1(2

1

)1(8

816328163

))2/tanh(()1(2

116

2

828)(

2

2

4222222

222213

C

Ca

ABABCBBCACBb

Ca

Cb

a

VbBbAbBCu


,))2/coth(()1(2

1

)1(8

816328163

))2/coth(()1(2

116

2

828)(

2

2

4222222

222223

C

Ca

ABABCBBCACBb

Ca

Cb

a

VbBbAbBCu

199

.)(sec)(tanh()1(2

1

)1(8

816328163

)(sec)(tanh()1(2

116

2

828)(

2

2

4222222

222233

hiC

Ca

ABABCBBCACBb

hiCa

Cb

a

VbBbAbBCu

The other exact solutions of the family are omitted for the sake of simplicity.


,))2/tan(()1(2

1

)1(8

816328163

))2/tan(()1(2

116

2

828)(

2

2

4222222

2222123

C

Ca

ABABCBBCACBb

Ca

Cb

a

VbBbAbBCu

Fig.6.19. Solution 123u represents the exact singular periodic traveling wave solution for

.1,3,1,2,1,1 VbaCBA

200

,))2/cot(()1(2

1

)1(8

816328163

))2/cot(()1(2

116

2

828)(

2

2

4222222

2222133

C

Ca

ABABCBBCACBb

Ca

Cb

a

VbBbAbBCu

.)(sec)(tan()1(2

1

)1(8

816328163

)(sec)(tan()1(2

116

2

828)(

2

2

4222222

2222143

C

Ca

ABABCBBCACBb

Ca

Cb

a

VbBbAbBCu


,

)1(

1

)1(2)1(8

816328163

)1(

1

)1(2

16

2

828)(

2

22

4222222

2

2

222253

cCC

A

Ca

ABABCBBCACBb

cCC

A

a

Cb

a

VbBbAbBCu


Other exact solutions of (6.19) are omitted for convenience.

The value of v (which are not shown here for the sake of simplicity) can be

premeditated by substituting the values of u in the first equation of the system (6.21).

201

Example 6.2.4. The (3 + 1)-Dimensional Burgers Equations

Now consider the (3 + 1)-dimensional Burgers equations [219] in the form,

,

,

,

yz

yx

zzyyxxxxxt

wu

vu

uuucwubvuauuu

(6.30)

where cba and, are nonzero constants. The (3 + 1)-dimensional Burgers equations crop

up in various areas of applied mathematics, such as modeling of gas dynamics and traffic

flow.

The wave variable Vtzyxuu ),( converts the system (6.30) into a system of

ordinary differential equations as

,

,

,3

wu

vu

uucwubvuauVu

(6.31)

where prime stands for ordinary derivatives with respect to . Integrating the last two

equations of system (6.31) once, gives ,wvu and constants of integration are taken

as zero. The first equation in the system (6.31) after integrating once becomes:

.032

12

Cuu

cbaVu (6.32)

where 1C is an integration constant. Now, considering the homogeneous balance between

u and 2u in (6.32), we deduce that 12 mm . i.e. 1m . Therefore, the trial solution

becomes,

)()()( 10

1

1

kku . (6.33)

Proceeding as before the following three solution sets are obtained:

Set 1.

,)1(6

1cba

C

,00 ,01 ,kk 0)()1(63 cbaCkAV ,

,)3612121236

2666223612

121236367236()(2

1

20

22220

220

2000

20000

20

20

20

002

1

cCkbackakbkACk

abAcAbAabccakckC

akCbkCBkACkBCcba

C

(6.34)

202

where 0 , k, A, B and C are arbitrary constants.

Set 2.

,)(6 22

1cba

kAkCkB

00 , 01 , kk , 0)(663 cbakCkAV ,

,)3612121236

6662223612

121236367236()(2

1

2220

220

220

2000

00020

20

20

20

002

1

CkcbackbkakACk

AcAbAaabbccakckC

akCbkCBkACkBCcba

C

(6.35)

where 0 , k, A, B and C are arbitrary constants.

Set 3.

,)1(6

1cba

C

,

)1)((2

)44(3 2

1Ccba

BABC

,00 ,

)1(2 C

Ak

,)( 0cbaV

,)36222144144()(2

1 220

220

220

220

20

201 AcbacabcabBBC

cbaC

(6.36)

where 0 , A, B and C are arbitrary constants.

Substituting equations (6.34)-(6.36) in solution formula (6.33) yields:

,)/()1(6

)( 01 GGkcba

Cu

(6.37)

,)/()(6

)(1

22

02

GGd

cba

kAkCkBu (6.38)

,)/()1(2)1)((2

)44(3

)/()1(2

)1(6)(

12

03

GGC

A

Ccba

BABC

GGC

A

cba

Cu

(6.39)

where Vtzyx ; 0 , k, A, B and C are arbitrary constants.


of system (6.30) are obtained:

203


,)2

1tanh(

)1(2

1)1(6)( 0

11

A

Ck

cba

Cu

Fig. 6.20. Shows the form of the exact kink-type solution of 11u for ,2,1,1 CBA

,1,1 ak .,1,1,1 0 zyxcb Kink waves are traveling waves which take

place from one asymptotic state to another. The kink solution approaches a constant at infinity.

,)2

1(cot

)1(2

1)1(6)( 0

21

hA

Ck

cba

Cu

,)(sec)(tan)1(2

1)1(6)( 0

31

hihA

Ck

cba

Cu

,)(csc)(cot)1(2

1)1(6)( 0

41

hhA

Ck

cba

Cu

,)4

1coth()

4

1(tan2

)1(4

1)1(6)( 0

51

hA

Ck

cba

Cu

,)sinh(

)(cos)(

)1(2

1)1(6)(

22

061

HF

hFHFA

Ck

cba

Cu

204

,)sinh(

)(cos)(

)1(2

1)1(6)(

22

071

HF

hFHFA

Ck

cba

Cu


,

)2

1cosh()

2

1sinh(

)2

1(cos2

)1(6)( 0

81

A

hBk

cba

Cu

,

)2

1sinh()

2

1cosh(

)2

1(sin2

)1(6)( 0

91

A

hBk

cba

Cu

,)cosh()sinh(

)(cos2)1(6)( 0

101

iA

hBk

cba

Cu

.)sinh()cosh(

)(sin2)1(6)( 0

111

A

hBk

cba

Cu


,)2

1tan(

)1(2

1)1(6)( 0

121

A

Ck

cba

Cu


.,1,1,1,1,1,2,1,1 0 zyxcbakCBA The traveling wave

solutions which are periodic in nature like )cos( tx are called periodic solutions.

205

,)2

1(cot

)1(2

1)1(6)( 0

131

A

Ck

cba

Cu

,)sec()tan()1(2

1)1(6)( 0

141

A

Ck

cba

Cu

,)csc)cot()1(2

1)1(6)( 0

151

A

Ck

cba

Cu

,)4

1cot()

4

1tan(2

)1(4

1)1(6)( 0

161

A

Ck

cba

Cu

,)sin(

)cos()(

)1(2

1)1(6)(

22

0171

HF

FHFA

Ck

cba

Cu

,)sin(

)cos()(

)1(2

1)1(6)(

22

0181

HF

FHFA

Ck

cba

Cu

where F and H are arbitrary constants such that 022 HF .

,

)2

1cos()

2

1sin(

)2

1(cos2

)1(6)( 0

191

A

Bk

cba

Cu

,

)2

1sin()

2

1cos(

)2

1sin(2

)1(6)( 0

201

A

Bk

cba

Cu

,)cos()sin(

)(cos2)1(6)( 0

211

A

Bk

cba

Cu

.

)2

1sin()

2

1cos(

)2

1sin(2

)1(6)( 0

221

A

Bk

cba

Cu


,

)sinh()cosh()1(

)1(6)(

1

10

231

AAcC

Ack

cba

Cu

206

,)sinh()cosh()1(

)sinh()cosh()1(6)(

1

0241

AAcC

AAAk

cba

Cu


When 0)1( C and 0 BA , the solution of system (6.30) is

,)1(

1)1(6)(

2

0251

cCk

cba

Cu


Surrogating the solutions )(G of (1.49) in (6.38) and simplifying, the solutions of

system (6.30) are:

,)2

1tanh(

)1(2

1)(6)(

122

012

A

Ck

cba

kAkCkBu

where Vtzyx ; 0 , k, A, B and C are arbitrary constants.

,)2

1coth(

)1(2

1)(6)(

122

022

A

Ck

cba

kAkCkBu

.)(sec)tanh()1(2

1)(6)(

122

032

hiA

Ck

cba

kAkCkBu

The other solutions of system (6.30) are omitted for convenience.


,)2

1tan(

)1(2

1)(6)(

122

0122

A

Ck

cba

kAkCkBu

,)2

1cot(

)1(2

1)(6)(

122

0132

A

Ck

cba

kAkCkBu

.)sec()tan()1(2

1)(6)(

122

0142

A

Ck

cba

kAkCkBu

207


,)1(2

1)(6)(

1

2

22

0252

cCk

cba

kAkCkBu


Lastly, surrogating the solutions )(G of (1.49) into (6.39) and simplifying, the solutions

of system (6.30) are:


,)2

1tanh(

)1)((4

)44(3)

2

1tanh(

3)(

1

2

2

013

Ccba

BABC

cbau

where Vtzyx ; 0 , A, B and C are arbitrary constants.

,)2

1coth(

)1)((4

)44(3)

2

1coth(

3)(

1

2

2

023

Ccba

BABC

cbau

,)(sec)tanh()1)((4

)44(3

)(sec)tanh(3

)(

1

2

2

033

hiCcba

BABC

hicba

u

The other exact solutions are omitted for the sake of simplicity.


,)2

1tan(

)1)((4

)44(3)

2

1tan(

3)(

1

2

2

0123

Ccba

BABC

cbau

,)2

1cot(

)1)((4

)44(3)

2

1cot(

3)(

1

2

2

0133

Ccba

BABC

cbau

.)sec()tan()1)((4

)44(3

)sec()tan(3

)(

1

2

2

0143

Ccba

BABC

cbau

208


,1

2)1)((2

)44(31

2

6)(

1

22

2

2

0253

c

A

Ccba

BABC

c

A

cbau


Example 6.2.5. The (3 + 1) Dimensional Modified KdV-ZK Equation

Finally, consider the (3 + 1)-dimensional modified KdV-ZK equation [232] in the form

,02 zzxyyxxxxxt uuuuuu (6.40)

Now, using the wave transformation tVzyxutzyxu ),(),,,( that permits to

convert (6.40) in an ordinary differential equation as

,032 uuuuV (6.41)

where prime denotes ordinary derivatives with respect to . Integrating (6.41) once with

respect yields:

,033

1 31 uuVuC (6.42)

where 1C is an integral constant which is to be determined later. Now, cconsidering the

homogeneous balance between u and 3u in (6.42), we deduce that .23 mm i.e.

.1m Therefore, the trial solution becomes,

.)()()( 10

1

1

kku (6.43)

Proceeding as before the following four solution sets are obtained.

Set 1.

,

2

223,

2

)1(601

CkkAiCi

,01 ,kk

,0,2

366 1

2 CABBCV (6.44)

where k, , A, B and C are arbitrary constants.

209

Set 2.

,0,2

366

,,2

6,

2

223,0

12

22

101

CABBCV

kkkBCkAkiCkkAi

(6.45)


Set 3.

,0,446

,12

,122

443,0,

2

)1(6

12

2

101

CBBCAV

C

Ak

C

BBCAiCi

(6.46)

where , A, B and C are arbitrary constants.

Set 4.

,122

3,

2

423,

2

)1(6 22

101

C

kBCkAkiCkkAiCi

,2

312181812181836, 22222 ABAkkBCACkkCCkVkk (6.47)

,

2

122136 2

1

BAkkCAkCCiC


Substituting equations (6.44)-(6.47) into (6.43), the solution becomes

,)/(2

)1(6

2

223)(1 GGk

CiCkkAiu

(6.48)

where .2

366 2 tABBCzyx

,)/(2

6

2

223)(

122

2

GGdkBCkAkiCkkAi

u

(6.49)

where .2

366 2 tABBCzyx

,)/()1(2122

443)/(

)1(22

)1(6)(

12

3

GG

C

A

C

BBCAiGG

C

ACiu

(6.50)

210

where .446 2 tBBCAzyx

,)/(122

3

)/(2

)1(6

2

423)(

122

4

GGkC

kBCkAki

GGkCiCkkAi

u

(6.51)

where .2

312181812181836 22222 tABAkkBCACkkCCkzyx


of (6.40) are obtained:


,)

2

1tanh(

)1(2

1

2

)1(6

2

223)(1

1

A

Ck

CiCkkAiu

where .2

366 2 tABBCzyx

,)

2

1(cot

)1(2

1

2

)1(6

2

223)(2

1

hA

Ck

CiCkkAiu

Fig.6.22.Solution 21u explicates the soliton for ,0,9,2,5.0,1,3 ykCBA

.0z The particular types of solitary waves are solitons. The soliton solution is a spatially confined solution; hence 0)(),(),( uuu as , .tVx An

amazing chattel of solitons is that it maintains its individuality upon interacting with other solitons.

211

,)(sec)(tan)1(2

1

2

)1(6

2

223)(3

1

hihAC

kCiCkkAi

u

,)(csc)(cot)1(2

1

2

)1(6

2

223)(4

1

hhAC

kCiCkkAi

u

,)

4

1coth()

4

1(tan2

)1(4

1

2

)1(6

2

223)(5

1

hA

Ck

CiCkkAiu

,)sinh(

)(cos)(

)1(2

1

2

)1(6

2

223)(

22

61

HF

hFHFA

Ck

CiCkkAiu

,

)sinh(

)(cos)(

)1(2

1

2

)1(6

2

223)(

2271

HF

hFHFA

Ck

CiCkkAiu


,

)2

1cosh()

2

1sinh(

)2

1(cos2

2

)1(6

2

223)(8

1

A

hBk

CiCkkAiu

,

)2

1sinh()

2

1cosh(

)2

1(sin2

2

)1(6

2

223)(9

1

A

hBk

CiCkkAiu

,

)cosh()sinh(

)(cos2

2

)1(6

2

223)(10

1

iA

hBk

CiCkkAiu

.

)sinh()cosh(

)(sin2

2

)1(6

2

223)(11

1

A

hBk

CiCkkAiu


,)

2

1tan(

)1(2

1

2

)1(6

2

223)(12

1

A

Ck

CiCkkAiu

212

Fig.6.23. 121u represents the exact periodic traveling wave solution for ,5.1,2 BA

,)

2

1(cot

)1(2

1

2

)1(6

2

223)(13

1

A

Ck

CiCkkAiu

Fig. 6.24. Solution 131u explains the soliton for ,0,9,2,5.0,5.1,2 ykCBA

.0z Solitons are particular kinds of solitary waves. An amazing chattel of solitons is that it maintains its individuality upon interacting with other solitons.

,)sec()tan()1(2

1

2

)1(6

2

223)(14

1

AC

kCiCkkAi

u

,)csc)cot()1(2

1

2

)1(6

2

223)(15

1

AC

kCiCkkAi

u

213

,)

4

1cot()

4

1tan(2

)1(4

1

2

)1(6

2

223)(16

1

A

Ck

CiCkkAiu

,

)sin(

)cos()(

)1(2

1

2

)1(6

2

223)(

22

171

HF

FHFA

Ck

CiCkkAiu

,

)sin(

)cos()(

)1(2

1

2

)1(6

2

223)(

22

181

HF

FHFA

Ck

CiCkkAiu

where F and H are arbitrary constants such that 022 HF .

,

)2

1cos()

2

1sin(

)2

1(cos2

2

)1(6

2

223)(19

1

A

Bk

CiCkkAiu

,

)2

1sin()

2

1cos(

)2

1sin(2

2

)1(6

2

223)(20

1

A

Bk

CiCkkAiu

,

)cos()sin(

)(cos2

2

)1(6

2

223)(21

1

A

Bk

CiCkkAiu

.

)2

1sin()

2

1cos(

)2

1sin(2

2

)1(6

2

223)(22

1

A

Bk

CiCkkAiu


,)sinh()cosh()1(2

)1(6

2

223)(

1

1231

AAcC

Ack

CiCkkAiu

,)sinh()cosh()1(

)sinh()cosh(

2

)1(6

2

223)(

1

241

AAcC

AAAk

CiCkkAiu



,

)1(

1

2

)1(6

2

223)(

2

251

Cck

CiCkkAiu


Similarly, surrogating the solutions )(G of the (1.43) in (6.49) and simplifying, the

solutions of (6.40) are:

214


,)

2

1tanh(

)1(2

1

2

6

2

223)(

12212

A

Ck

kBCkAkiCkkAiu

where .2

366 2 tABBCzyx

,)

2

1coth(

)1(2

1

2

6

2

223)(

12222

A

Ck

kBCkAkiCkkAiu

.)(sec)tanh()1(2

1

2

6

2

223)(

12232

hiA

Ck

kBCkAkiCkkAiu

The other exact solutions of (6.40) are omitted for the sake of convenience.


,)

2

1tan(

)1(2

1

2

6

2

223)(

122122

A

Ck

kBCkAkiCkkAiu


,5.1,2 BA ,0C .0,0,9,1 zyk

215

,)

2

1cot(

)1(2

1

2

6

2

223)(

122132

A

Ck

kBCkAkiCkkAiu

,)sec()tan()1(2

1

2

6

2

223)(

122142

AC

kkBCkAkiCkkAi

u


,

)1(

1

2

6

2

223)(

1

2

22252

Cck

kBCkAkiCkkAiu


Similarly, surrogating the solutions )(G of (1.49) in (6.50) and simplifying, we obtain

the following solutions of (6.40):


,)2

1tanh(

122

443)

2

1tanh(

2

)1(6)(

1213

C

BBCAiCiu

where .446 2 tBBCAzyx

,)2

1coth(

122

443)

2

1coth(

2

)1(6)(

1223

C

BBCAiCiu

.)(sec)tanh(

122

443)(sec)tanh(

2

)1(6)(

1

233

hi

C

BBCAihi

Ciu



,)2

1tan(

122

443)

2

1tan(

2

)1(6)(

12123

C

BBCAiCiu

,)2

1cot(

122

443)

2

1cot(

2

)1(6)(

12133

C

BBCAiCiu

216

.)sec()tan(

122

443)sec()tan(

2

)1(6)(

1

2143

C

BBCAiCiu


,

)1(

1

12122

443

1

1

122

)1(6)(

1

2

2

2

253

CcC

A

C

BBCAi

cCC

ACiu


Lastly, surrogating the solutions )(G of (1.49) in (6.51) and simplifying, the following

solutions of (6.40) are obtained:


,)2

1tanh(

)1(2

1

122

3

)2

1tanh(

)1(2

1

2

)1(6

2

423)(

122

14

AC

kC

kBCkAki

AC

kCiCkkAi

u

where .2

312181812181836 22222 tABAkkBCACkkCCkzyx

217

,)2

1coth(

)1(2

1

122

3

)2

1coth(

)1(2

1

2

)1(6

2

423)(

122

24

AC

kC

kBCkAki

AC

kCiCkkAi

u

.)(sec)tanh()1(2

1

122

3

)(sec)tanh()1(2

1

2

)1(6

2

423)(

122

34

hiAC

kC

kBCkAki

hiAC

kCiCkkAi

u



,)2

1tan(

)1(2

1

122

3

)2

1tan(

)1(2

1

2

)1(6

2

423)(

122

124

AC

kC

kBCkAki

AC

kCiCkkAi

u

Fig.6.27. Solution 124u represents the periodic traveling wave solution for

,1,1 BA ,2C .0,0,9,2 zyk

218

,)2

1cot(

)1(2

1

122

3

)2

1cot(

)1(2

1

2

)1(6

2

423)(

122

134

AC

kC

kBCkAki

AC

kCiCkkAi

u

.)sec()tan()1(2

1

122

3

)sec()tan()1(2

1

2

)1(6

2

423)(

122

144

AC

kC

kBCkAki

AC

k

CiCkkAiu


,)1(

1

122

3

)1(

1

2

)1(6

2

423)(

1

2

22

2

254

Cck

C

kBCkAki

Cck

CiCkkAiu


Discussion

Copious precise traveling wave solutions of a variety of nonlinear evolution equations

have been constructed by applying novel )/( GG -expansion method. The obtained

solutions are more general and many known solutions are only a special case of them,

when parameters are given some special values. Moreover, the novel )/( GG -expansion

method is relatively resourceful and virtually suitable in the discovery of exact solutions

of nonlinear evolution equations.

219

Chapter 7

Novel )/( GG -Expansion Method for Fractional Partial

Differential Equations

220

7.1. Introduction

Differential equations of non integer order are generalizations of conventional differential

equations of integer order. Exploration and applications of integrals and derivatives of

arbitrary order are efficiently dealt with the field of mathematical analysis, called as

fractional calculus, which has engrossed in considerable interest in many disciplines, now

a day. The behavior of many physical systems can be perfectly defined by the fractional

theory. In recent years, we cannot rebuff the importance of fractional differential

equations because of their numerous applications in the areas of physics and engineering.

Many important phenomena in electro-magnetic, acoustics, visco-elasticity, electro-

chemistry and material science are better described by differential equations of non

integer order. For example, the nonlinear fluctuation of earthquakes can be modeled with

the help of fractional derivatives and the fluid-dynamic traffic model with fractional

derivatives can eradicate the problems arising from the huge traffic flow [77, 164, 193,

229]. On account of development of the computer and its exact description of count-less

real-life problems, fractional calculus has touched the height of fame and success now a

day, although it was invented three centuries ago by Newton and Leibniz. Further

applications of differential equations of non integer order can be found in [155-163].

The exact solutions of nonlinear fractional partial differential equations have a great

importance in nonlinear sciences. In the past, many analytical and numerical methods

have been proposed to obtain solutions of nonlinear differential equations, such as

Variational Iteration [169], Differential Transform [182], Modified Adomian’s

Decomposition [7, 96, 100], Homotopy Perturbation [168] methods and so on. But most

of the developed techniques have their limitations coupled with some inbuilt deficiencies

like limited convergence, divergent results, linearization, discretization, unrealistic

assumptions, huge computational work and non-compatibility with the versatility of

physical problems.

In this chapter, Novel ( / )G G -expansion method is extended to find the soliton solutions

of non integer order partial differential equations [27, 201, 202]. In the proposed

algorithm the following complex transformation is used:

,, utxu

,11

0

t

Vx

L

221

The proposed technique has been applied on a wide range of nonlinear diversified

physical problems including, high-dimensional nonlinear evolution equations have been

taking the advantage of the proposed technique. The proposed scheme is having the

qualities of compatibility with the complexity of such problems is very user friendly and

numerical results are very fortifying.


In this section, novel ( / )G G -expansion method is extended to find the soliton solutions

of non integer order partial differential equations including, the time fractional simplified

modified Camassa-Holm (MCH) equation, the time fractional BBM-Burger’s equation

and the space-time fractional SRLW equation. Physical properties of several nonlinear

traveling wave solutions are examined by graphs which are obtained for various values

of .

7.2.1. The Time Fractional Simplified MCH Equation

Consider the following time fractional simplified modified Camassa-Holm equation

[201] in the form

,1,0,where,02 2 xtxxxt uuuuuD (7.1)

which is the variation of the equation

.0,where,02 2 xtxxxt uuuuu (7.2)

Equation (7.1) is converted into an ordinary differential equation of integer order by

using (1.6) and (1.52), and after integrating once yields:

,03

2 1

32 C

uLuVLuLV (7.3)

where 1C is an integral constant which is to be determined later.

Considering the homogeneous balance between u and 3u in (14), we obtain .23 mm

i.e. .1m Therefore the trial solution formula (1.54) becomes,

.)()()( 10

1

1

kku (7.4)

Using (7.4) in (7.3), left hand side is converted into polynomials in iGGk / and

,/i

GGk ).,,2,1,0( mi Equating the coefficients of same power of the resulted

222

polynomials to zero, a system of algebraic equations for 0 , 1 , 1 , k , 1C , L and V is

obtained. Solving the system, the following four solution sets are attained:

Set 1.

,

244

22622

0

BBCAL

CkkALi

,

244

16222

1

BBCAL

CLi

,

244

422

BBCAL

LV

kkLL , , ,01 ,01 C (7.5)

where k, L, A, B and C are arbitrary constants.

Set.2.

,

244

22622

0

BBCAL

CkkALi

,244

6222

22

1

BBCAL

BCkkkALi

,

244

422

BBCAL

LV

kkLL , , ,01 ,01 C (7.6)

where k, L, A, B and C are arbitrary constants.

Set 3.

,

1442

132

221

BBCAL

CLi

,114422

44322

2

1

CBBCAL

BBCALi

,

1442

222

BBCAL

LV

,

)1(2

C

Ak ,LL ,00 ,01 C (7.7)

where L, A, B and C are arbitrary constants

Set 4.

,12442

44622

2

1

CBBCAL

BBCALi

,

244

422

BBCAL

LV

,)1(2

C

Ak ,LL ,0,0 10 ,01 C (7.8)

where L, A, B and C are arbitrary constants.

Substituting equations (7.5)-(7.8) in (7.4), the solution formula (7.4) becomes,

223

,)/(

244

162

244

226)(

22221 GGk

BBCAL

CLi

BBCAL

CkkALiu

(7.9)

where

.1244

422

t

BBCAL

LLx

,)/(244

62

244

226)(

1

22

22

222

GGk

BBCAL

BCkkkALi

BBCAL

CkkALiu

(7.10)

where

.1244

422

t

BBCAL

LLx

,)/()1(2114422

443

)/()1(21442

132)(

1

22

2

223

GGC

A

CBBCAL

BBCALi

GGC

A

BBCAL

CLiu

(7.11)

where

.11442

222

t

BBCAL

LLx

,)/()1(212442

446)(

1

22

2

4

GG

C

A

CBBCAL

BBCALiu

(7.12)

where

.1244

422

t

BBCAL

LLx

Inserting the solutions )(G of (1.56) in (7.9) and simplifying, the solutions are:


,)2/tanh(

12

1

244

162

244

226)(

2222

11

AC

k

BBCAL

CLi

BBCAL

CkkALiu

224

Fig. 7.1 (a-d). Shows the form of the exact kink-type solution of 11u for different values

of and ,2,1,1 CBA .1,1,1,1 Lk Kink waves are traveling waves

which take place from one asymptotic state to another. The kink solution approaches a constant at infinity.

,)2/coth(

12

1

244

162

244

226)(

2222

21

AC

k

BBCAL

CLi

BBCAL

CkkALiu

225

Fig.7.2 (a-d): Singular kink solution of 21u for different values of and

,1,2,5 CBA .5,1,1,2 Lk

,)(sec)tanh(

12

1

244

162

244

226)(

2222

31

hiAC

k

BBCAL

CLi

BBCAL

CkkALiu

,)(csc)coth(

12

1

244

162

244

226)(

2222

41

hAC

k

BBCAL

CLi

BBCAL

CkkALiu

,)4/coth()4/tanh(2

14

1

244

162

244

226)(

2222

51

AC

k

BBCAL

CLi

BBCAL

CkkALiu

226

,

)sinh(

)cosh()(

12

1

244

162

244

226)(

22

2222

61

BF

FHFA

Ck

BBCAL

CLi

BBCAL

CkkALiu

,

)sinh(

)cosh()(

12

1

244

162

244

226)(

22

2222

71

BF

FHFA

Ck

BBCAL

CLi

BBCAL

CkkALiu


,)2/cosh()2/sinh(

)2/cosh(2

244

162

244

226)(

2222

81

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

,)2/sinh()2/cosh(

)2/sinh(2

244

162

244

226)(

2222

91

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

,)cosh()sinh(

)cosh(2

244

162

244

226)(

2222

101

iA

Bk

BBCAL

CLi

BBCAL

CkkALiu

.)sinh()cosh(

)sinh(2

244

162

244

226)(

2222

111

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

227


,)2/tan(

12

1

244

162

244

226)(

2222

121

AC

k

BBCAL

CLi

BBCAL

CkkALiu

,)2/cot(

12

1

244

162

244

226)(

2222

131

AC

k

BBCAL

CLi

BBCAL

CkkALiu

,)sec()tan(

12

1

244

162

244

226)(

2222

141

AC

k

BBCAL

CLi

BBCAL

CkkALiu

,)(csc)cot(

12

1

244

162

244

226)(

2222

151

hAC

k

BBCAL

CLi

BBCAL

CkkALiu

,

)sin(

)cos()(

12

1

244

162

244

226)(

22

2222

161

BF

FHFA

Ck

BBCAL

CLi

BBCAL

CkkALiu

,

)sin(

)cos()(

12

1

244

162

244

226)(

22

2222

171

BF

FHFA

Ck

BBCAL

CLi

BBCAL

CkkALiu

228

,

)sin(

)cos()(

12

1

244

162

244

226)(

22

2222

181

BF

FHFA

Ck

BBCAL

CLi

BBCAL

CkkALiu


,)2/cos()2/sin(

)2/cos(2

244

162

244

226)(

2222

191

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

,)2/sin()2/cos(

)2/sin(2

244

162

244

226)(

2222

201

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

,)cos()sin(

)cos(2

244

162

244

226)(

2222

211

A

Bk

BBCAL

CLi

BBCAL

CkkALiu

.)2/sin()2/cos(

)2/sin(2

244

162

244

226)(

2222

221

A

Bk

BBCAL

CLi

BBCAL

CkkALiu


,

)sinh()cosh()1(

244

162

244

226)(

1

1

2222

231

AAcC

Ack

BBCAL

CLi

BBCAL

CkkALiu

229

,

)sinh()cosh()1(

))sinh()(cosh(

244

162

244

226)(

1

2222

241

AAcC

AAAk

BBCAL

CLi

BBCAL

CkkALiu


When 0 BA and ,0)1( C the solution of (7.1) is

,)1(

1

244

162

244

226)(

2

2222

251

cCk

BBCAL

CLi

BBCAL

CkkALiu



are obtained:


,)2/tanh(

12

1

244

62

244

226)(

1

22

22

22

12

AC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu

,)2/coth(

12

1

244

62

244

226)(

1

22

22

22

22

AC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu

.))(sec)(tanh(

12

1

244

62

244

226)(

1

22

22

22

32

hiAC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu

230


,)2/tan(

12

1

244

62

244

226)(

1

22

22

22

122

AC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu

Fig.7.3 (a-d): Solution 122u represents the periodic traveling wave solution for different

values of and ,2,1,2 CBA .12,1,1,2 Lk

,)2/cot(

12

1

244

62

244

226)(

1

22

22

22

132

AC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu

231

.))sec()(tan(

12

1

244

62

244

226)(

1

22

22

22

142

AC

k

BBCAL

BCkkkALi

BBCAL

CkkALiu


,)1(

1

244

62

244

226)()(

1

2

22

22

222

252

cCk

BBCAL

BCkkkALi

BBCAL

CkkALiuu


Similarly by inserting the solutions )(G of (1.56) in (7.11), the solutions are


,))2/tanh(()1(2

1

114422

443

))2/tanh(()1(2

1

1442

132)(

1

22

2

22

13

CCBBCAL

BBCALi

CBBCAL

CLiu

,))2/coth((

)1(2

1

114422

443

))2/coth(()1(2

1

1442

132)(

1

22

2

22

23

CCBBCAL

BBCALi

CBBCAL

CLiu

.)(sec)(tanh()1(2

1

114422

443

)(sec)(tanh()1(2

1

1442

132)(

1

22

2

22

33

hiCCBBCAL

BBCALi

hiCBBCAL

CLiu


232

When 0442 BBCA and 0)1( CA (or 0)1( CB ), the solutions are:

,))2/tan((

)1(2

1

114422

443

))2/tan(()1(2

1

1442

132)(

1

22

2

22

123

CCBBCAL

BBCALi

CBBCAL

CLiu

Fig.7.4 (a-d): Illustrates the sketch of the singular periodic traveling wave solution of 123u

for different values of and ,2,1,1 CBA .9,1,1,2 Lk

,))2/cot((

)1(2

1

114422

443

))2/cot(()1(2

1

1442

132)(

1

22

2

22

133

CCBBCAL

BBCALi

CBBCAL

CLiu

233

.)(sec)(tan()1(2

1

114422

443

)(sec)(tan()1(2

1

1442

132)(

1

22

2

22

143

CCBBCAL

BBCALi

CBBCAL

CLiu


,

)1(

1

)1(2114422

443

)1(

1

)1(21442

132)(

1

222

2

222

253

cCC

A

CBBCAL

BBCALi

cCC

A

BBCAL

CLiu


Lastly, by surrogating the solutions )(G of (1.56) in (7.12), the solutions are:


,))2/tanh(()1(2

1

12442

446)(

1

22

214

CCBBCAL

BBCALiu

,))2/coth(()1(2

1

12442

446)(

1

22

224

CCBBCAL

BBCALiu

.)(sec)(tanh()1(2

1

12442

446)(

1

22

234

hi

CCBBCAL

BBCALiu



,))2/tan(()1(2

1

12442

446)(

1

22

2124

CCBBCAL

BBCALiu

,))2/cot(()1(2

1

12442

446)(

1

22

2134

CCBBCAL

BBCALiu

234

.)(sec)(tan()1(2

1

12442

446)(

1

22

2144

CCBBCAL

BBCALiu


,)1(

1

)1(212442

446)(

1

222

2254

cCC

A

CBBCAL

BBCALiu


7.2.2. The Time Fractional BBM-Burgers Equation

The BBM-Burgers equation [202] in time fractional form is

.10,0,02

2

t

uuuuD

x

xtxxt (7.13)

Utilizing (1.6) and (1.52) that converts (7.13) in an ordinary differential equation of

integer order and after integrating once yields,

,02

122 CuVLu

LuVL (7.14)

where 1C is an integration constant. Considering the homogeneous balance between

u and 2u in (7.14), we obtain .2m Therefore, the trial solution (1.54) becomes,

.)()()()()(2

210

1

1

2

2

kkkku (7.15)

Proceeding in a similar fashion as before, the following three solution sets are obtained:

Set 1.

,221121 AkkCCVL ,1122

2 CVL

kkLLVV ,, , ,01 ,02 (7.16)

,44442

1 22221 LVVLABCLVVLABC

LC

where k, L, V, A, B and C are arbitrary constants.

235

Set.2.

,

88112112 2222

0L

LVLABCBkCAkCV

,22112 21 AkCVLBkAkC

,11222

2 BkAkCVL kkLLVV ,, ,02 ,01 (7.17)

,44442

1 22221 LVLABCVLVLABCV

LC


Set 3.

,1122

2 CVL

,442 22

0L

LVABCVL

)1(2

C

Ak ,

,

14

4432

22

2C

ABCVL

,01 ,01 ,, LLVV (7.18)

,44

4444

448 2222

1

LVABCLV

LVABCLV

LC

where V, L, A, B and C are arbitrary constants.

Substituting equations (7.16)- (7.18) in (7.15), the trial solution (7.15) becomes,

,)/(112)/(22112

88112112)(

22

2222

1

GGkCVLGGkAkkCCVL

L

LVLABCBkCAkCVu

(7.19)

,)/(112

)/(22112

88112112)(

222

12

2222

2

GGkBkAkCVL

GGkAkCBkAkCVL

L

LVLABCBkCAkCVu

(7.20)

,)/()1(214

443

)/()1(2

112442

)(

2

2

22

2

222

3

GGC

A

C

ABCVL

GGC

ACVL

L

LVABCVLu

(7.21)

236

where

.1

tVLx

Inserting the solutions )(G of (1.56) in (7.19), the following solutions are obtained:


,)2/tanh(12

1112

)2/tanh(12

122112

88112112)(

2

2

222211

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

,)2/coth(12

1112

)2/coth(12

122112

88112112)(

2

2

222221

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

25.0)a( 50.0)b(

237

75.0)c( 0.1)d(

Fig. 7.5 (a-d): Exact singular soliton solution of 21u for different values of and

,2,1,1 CBA .1,1,1 VLk

,)(sec)tanh(12

1112

)(sec)tanh(12

122112

88112112)(

2

2

2222

31

hiAC

kCVL

hiAC

kAkkCCVL

L

LVLABCBkCAkCVu

,)(csc)coth(12

1112

)(csc)coth(12

122112

88112112)(

2

2

222241

hAC

kCVL

hAC

kAkkCCVL

L

LVLABCBkCAkCVu


1112

)4/coth()4/tanh(214

122112

88112112)(

2

2

222251

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

238

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

122112

88112112)(

222

2

22

222261

BF

FHFA

CkCVL

BF

FHFA

CkAkkCCVL

L

LVLABCBkCAkCVu

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

122112

88112112)(

222

2

222

222271

BF

FHFA

CkCVL

BF

FHFA

CkAkkCCVL

L

LVLABCBkCAkCVu


,)2/cosh()2/sinh(

)2/cosh(2112

)2/cosh()2/sinh(

)2/cosh(222112

88112112)(

2

2

222281

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu

,)2/sinh()2/cosh(

)2/sinh(2112

)2/sinh()2/cosh(

)2/sinh(222112

88112112)(

2

2

222291

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu

239

,)cosh()sinh(

)cosh(2112

)cosh()sinh(

)cosh(222112

88112112)(

2

2

2222101

iA

BkCVL

iA

BkAkkCCVL

L

LVLABCBkCAkCVu

.)sinh()cosh(

)sinh(2112

)sinh()cosh(

)sinh(222112

88112112)(

2

2

2222111

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu


,)2/tan(12

1112

)2/tan(12

122112

88112112)(

2

2

2222

121

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

,)2/cot(12

1112

)2/cot(12

122112

88112112)(

2

2

2222131

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

,)sec()tan(12

1112

)sec()tan(12

122112

88112112)(

2

2

2222141

AC

kCVL

AC

kAkkCCVL

L

LVLABCBkCAkCVu

240

,)(csc)cot(12

1112

)(csc)cot(12

122112

88112112)(

2

2

2222151

hAC

kCVL

hAC

kAkkCCVL

L

LVLABCBkCAkCVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

122112

88112112)(

222

2

22

2222

161

BF

FHFA

CkCVL

BF

FHFA

CkAkkCCVL

L

LVLABCBkCAkCVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

122112

88112112)(

222

2

22

2222

171

BF

FHFA

CkCVL

BF

FHFA

CkAkkCCVL

L

LVLABCBkCAkCVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

122112

88112112)(

222

2

22

2222181

BF

FHFA

CkCVL

BF

FHFA

CkAkkCCVL

L

LVLABCBkCAkCVu


241

,)2/cos()2/sin(

)2/cos(2112

)2/cos()2/sin(

)2/cos(222112

88112112)(

2

2

2222

191

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu

,)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(222112

88112112)(

2

2

2222

201

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu

,)cos()sin(

)cos(2112

)cos()sin(

)cos(222112

88112112)(

2

2

2222

211

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu

.)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(222112

88112112)(

2

2

2222

221

A

BkCVL

A

BkAkkCCVL

L

LVLABCBkCAkCVu


,)sinh()cosh()1(

112


88112112)(

2

1

12

1

1

2222231

AAcC

AckCVL

AAcC

AckAkkCCVL

L

LVLABCBkCAkCVu

242

Fig.7.6 (a-d): Solution 231u represents the bell-shaped 2sec h solitary traveling wave

solution for different values of and ,1,1,1,2,0,1 VLkCBA estranged by

infinite wings or infinite tails.

243

,)sinh()cosh()1(

))sinh()(cosh(112

)sinh()cosh()1(

))sinh()(cosh(22112

88112112)(

2

1

2

1

2222241

AAcC

AAAkCVL

AAcC

AAAkAkkCCVL

L

LVLABCBkCAkCVu



,)1(

1112

)1(

122112

88112112)(

2

2

2

2

2222251

cCkCVL

cCkAkkCCVL

L

LVLABCBkCAkCVu


Inserting the solutions )(G of (1.56) in (7.20), the solutions are:


,)2/tanh(12

1112

)2/tanh(12

122112

88112112)(

222

1

2

222212

AC

kBkAkCVL

AC

kAkCBkAkCVL

L

LVLABCBkCAkCVu

,)2/coth(12

1112

)2/coth(12

122112

88112112)(

222

1

2

222222

AC

kBkAkCVL

AC

kAkCBkAkCVL

L

LVLABCBkCAkCVu

244

.))(sec)(tanh(12

1112

))(sec)(tanh(12

1

22112

88112112)(

222

1

2

2222

32

hiAC

kBkAkCVL

hiAC

k

AkCBkAkCVL

L

LVLABCBkCAkCVu


,)2/tan(12

1112

)2/tan(12

122112

88112112)(

222

1

2

2222

122

AC

kBkAkCVL

AC

kAkCBkAkCVL

L

LVLABCBkCAkCVu

,)2/cot(12

1112

)2/cot(12

122112

88112112)(

222

1

2

2222132

AC

kBkAkCVL

AC

kAkCBkAkCVL

L

LVLABCBkCAkCVu

.))sec()(tan(12

1112

))sec()(tan(12

1

22112

88112112)(

222

1

2

2222

142

AC

kBkAkCVL

AC

k

AkCBkAkCVL

L

LVLABCBkCAkCVu

245


,)1(

1112

)1(

122112

88112112)(

2

2

22

1

2

2

2222

252

cCkBkAkCVL

cCkAkCBkAkCVL

L

LVLABCBkCAkCVu


Lastly, inserting the solutions )(G of (1.56) in (7.21) and simplifying, the solutions are:


,))2/tanh(()1(2

1

14

443

))2/tanh(()1(2

1112

442)(

2

2

22

2

222

13

CC

ABCVL

CCVL

L

LVABCVLu

,))2/coth(()1(2

1

14

443

))2/coth(()1(2

1112

442)(

2

2

22

2

222

23

CC

ABCVL

CCVL

L

LVABCVLu

.)(sec)(tanh()1(2

1

14

443

)(sec)(tanh()1(2

1

112442

)(

2

2

22

2

222

33

hiCC

ABCVL

hiC

CVLL

LVABCVLu


246


,))2/tan(()1(2

1

14

443

))2/tan(()1(2

1112

442)(

2

2

22

2

222

123

CC

ABCVL

CCVL

L

LVABCVLu

Fig.7.7 (a-d): Solution 123u represents the periodic traveling wave solution for

different values of and .1,1,2,2,1,2 VLkCBA

,))2/cot(()1(2

1

14

443

))2/cot(()1(2

1112

442)(

2

2

22

2

222

133

CC

ABCVL

CCVL

L

LVABCVLu

247

.)(sec)(tan()1(2

1

14

443

)(sec)(tan()1(2

1112

442)(

2

2

22

2

222

143

CC

ABCVL

CCVL

L

LVABCLVu

When 0)1( C and 0 BA , the solution of (7.13) is,

,)1(

1

)1(214

443

)1(

1

)1(2112

442)(

2

22

22

2

2

222

253

cCC

A

C

ABCVL

cCC

ACVL

L

LVABCVLu


The other exact solutions of (7.13) are omitted here for convenience.

7.2.3. The Space-Time Fractional SRLW Equation

Consider the following space-time fractional SRLW equation [27] in the form

,10,02222 uDDuDuDuDuDuDuD xttxxtxt (7.22)

which arises in several physical applications including ion sound waves in plasma.

Using equations (1.6) and (1.52) that convert (7.22) in an ordinary differential equation of

integer order and after integrating twice yields,

,02

11

22222 CuVLuLVuVL (7.23)

where 1C is an integration constant. Considering the homogeneous balance between

u and 2u in (7.23), we obtain .22 mm i.e. .2m Therefore, the trial solution (1.54)

becomes,

.)()()()()(2

210

1

1

2

2

kkkku (7.24)

Proceeding as before, the following three solution sets are obtained:

Set 1.

,1122

2 CVL ,24212 21 kAACCkkCVL

,882412121212 222220

L

V

V

LABBCCkkACkAkkCLV

248

kkLLVV ,, , ,01 ,02 (7.25)

,22

328816162

332422222

33

1L

V

V

LLVCBABCABABCB

VLC


Set 2.

,882412121212 222220

L

V

V

LABBCCkkACkAkkCLV

,243322212 32223321 BCkCkACkkAAkkABkCBkLV

,22222212 243223422422 BCkABkCkACkkAAkkBBkkCLV

kkLLVV ,, ,02 ,01 (7.26)

,22

1328816162

332422222

33

1V

L

L

VLVCBABCABABCB

VLC

where k, L, V, A, B and C are arbitrary constants

Set 3.

,1122

2 CLV ,828 20

L

V

V

LBABCLV ,

)1(2

C

Ak

,81632816)1(4

3 4222222

22 ABABCBBCACBC

LV

,01 ,01 ,, LLVV (7.27)

,22

)64642568128128(33

2224222331

V

L

L

V

LVBABCACBABCBVLC

where V, L, A, B and C are arbitrary constants.

Substituting equations (7.25)- (7.27) in (7.24), The solution takes the form,

,)/(112

)/(24212

882412121212)(

22

2

222221

GGkCVL

GGkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

(7.28)

249

,)/()222

222(12)/(

243322212

882412121212)(

224

3223422421

3222332

222222

GGkBCkABkCk

ACkkAAkkBBkkCLVGGk

BCkCkACkkAAkkABkCBkLV

L

V

V

LABBCCkkACkAkkCLVu

(7.29)

,)/()1(2

81632816)1(4

3

)/()1(2

112828)(

2

4222222

2

2

223

GGC

A

ABABCBBCACBC

LV

GGC

ACLV

L

V

V

LBABCLVu

(7.30)

where

;11

0

t

Vx

L k, L, V, A, B and C are arbitrary constants.


are obtained:


,)2/tanh(12

1112

)2/tanh(12

124212

882412121212)(

2

2

2

2222211

AC

kCVL

AC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)2/coth(12

1112

)2/coth(12

124212

882412121212)(

2

2

2

2222221

AC

kCVL

AC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

250

,)(sec)tanh(12

1112

)(sec)tanh(12

124212

882412121212)(

2

2

2

2222231

hiAC

kCVL

hiAC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)(csc)coth(12

1112

)(csc)coth(12

124212

882412121212)(

2

2

2

2222241

hAC

kCVL

hAC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu


1112

)4/coth()4/tanh(214

1

24212

882412121212)(

2

2

2

2222251

AC

kCVL

AC

k

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

1

24212

882412121212)(

222

2

22

2

2222261

BF

FHFA

CkCVL

BF

FHFA

Ck

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

251

,)sinh(

)cosh()(

12

1112

)sinh(

)cosh()(

12

1

24212

882412121212)(

222

2

22

2

2222271

BF

FHFA

CkCVL

BF

FHFA

Ck

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu


,)2/cosh()2/sinh(

)2/cosh(2112

)2/cosh()2/sinh(

)2/cosh(224212

882412121212)(

2

2

2

2222281

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)2/sinh()2/cosh(

)2/sinh(2112

)2/sinh()2/cosh(

)2/sinh(224212

882412121212)(

2

2

2

2222291

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)cosh()sinh(

)cosh(2112

)cosh()sinh(

)cosh(224212

882412121212)(

2

2

2

22222101

iA

BkCVL

iA

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

252

.)sinh()cosh(

)sinh(2112

)sinh()cosh(

)sinh(224212

882412121212)(

2

2

2

22222111

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu


,)2/tan(12

1112

)2/tan(12

124212

882412121212)(

2

2

2

22222121

AC

kCVL

AC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)2/cot(12

1112

)2/cot(12

124212

882412121212)(

2

2

2

22222131

AC

kCVL

AC

kkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)sec()tan(12

1112

)sec()tan(12

1

24212

882412121212)(

2

2

2

22222141

AC

kCVL

AC

k

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

253

,)(csc)cot(12

1112

)(csc)cot(12

1

24212

882412121212)(

2

2

2

22222151

hAC

kCVL

hAC

k

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

882412121212)(

222

2

22

2

22222161

BF

FHFA

CkCVL

BF

FHFA

Ck

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

882412121212)(

222

2

22

2

22222171

BF

FHFA

CkCVL

BF

FHFA

Ck

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)sin(

)cos()(

12

1112

)sin(

)cos()(

12

1

24212

882412121212)(

222

2

22

2

22222181

BF

FHFA

CkCVL

BF

FHFA

Ck

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu


254

,)2/cos()2/sin(

)2/cos(2112

)2/cos()2/sin(

)2/cos(224212

882412121212)(

2

2

2

22222191

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(224212

882412121212)(

2

2

2

22222201

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

,)cos()sin(

)cos(2112

)cos()sin(

)cos(224212

882412121212)(

2

2

2

22222211

A

BkCVL

A

BkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

.)2/sin()2/cos(

)2/sin(2112

)2/sin()2/cos(

)2/sin(2

24212

882412121212)(

2

2

2

22222221

A

BkCVL

A

Bk

kAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

255


,)sinh()cosh()1(

112


882412121212)(

2

1

12

1

12

22222231

AAcC

AckCVL

AAcC

AckkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu

25.0)a( 50.0)b(

75.0)c( 0.1)d(

Fig.7.8 (a-d): Solution 231u symbolizes the bell-shaped 2sec h solitary traveling wave

solution for different values of and ,1,1,1,2,0,1 VLkCBA alienated by

infinite wings or infinite tails.

256

,)sinh()cosh()1(

))sinh()(cosh(112

)sinh()cosh()1(

))sinh()(cosh(24212

882412121212)(

2

1

2

1

2

22222241

AAcC

AAAkCVL

AAcC

AAAkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu



,)1(

1112

)1(

124212

882412121212)(

2

2

2

2

2

22222251

cCkCVL

cCkkAACCkkCVL

L

V

V

LABBCCkkACkAkkCLVu


Inserting the solutions )(G of (1.56) in (7.29) and simplifying, we obtain the following

solutions:


,)2/tanh(

12

1)2222

22(12)2/tanh(12

1

243322212

882412121212)(

2

24322

342242

1

3222332

2222212

AC

kBCkABkCkACkkA

AkkBBkkCLVAC

k


L

V

V

LABBCCkkACkAkkCLVu

where

;11

0

VtLx

k, L, V, A, B and C are arbitrary constants.

257

,)2/coth(

12

1)2222

22(12)2/coth(12

1

243322212

882412121212)(

2

24322

342242

1

3222332

2222222

AC

kBCkABkCkACkkA

AkkBBkkCLVAC

k


L

V

V

LABBCCkkACkAkkCLVu

.))(sec)(tanh(

12

1

)22222

2(12))(sec)(tanh(12

1

243322212

882412121212)(

2

24322342

242

1

3222332

2222232

hiAC

k

BCkABkCkACkkAAkkB

BkkCLVhiAC

k


L

V

V

LABBCCkkACkAkkCLVu

The other exact solutions of (7.22) are omitted for convenience.


,)2/tan(

12

1)2222

22(12)2/tan(12

1

243322212

882412121212)(

2

24322

342242

1

3222332

22222122

AC

kBCkABkCkACkkA

AkkBBkkCLVAC

k


L

V

V

LABBCCkkACkAkkCLVu

258

25.0)a( 50.0)b(

75.0)c( 0.1)d(

Fig.7.9 (a-d): Solution 122u corresponds to the exact periodic traveling wave solution for

different values of and ,2,1,2 CBA .1,1,1 VLk

259

,)2/cot(

12

1)2222

22(12)2/cot(12

1

243322212

882412121212)(

2

24322

342242

1

3222332

22222132

AC

kBCkABkCkACkkA

AkkBBkkCLVAC

k


L

V

V

LABBCCkkACkAkkCLVu

.))sec()(tan(

12

1

)222222

(12))sec()(tan(12

1

243322212

882412121212)(

2

243223422

42

1

3222332

22222142

AC

k

BCkABkCkACkkAAkkBBk

kCLVAC

k


L

V

V

LABBCCkkACkAkkCLVu


,)1(

1)222

222(12)1(

1

)2433222(12

882412121212)(

2

2

24

322342242

1

2

3222332

22222252

cCkBCkABkCk

ACkkAAkkBBkkCLVcC

k

BCkCkACkkAAkkABkCBkLVL

V

V

L

ABBCCkkACkAkkCLVu


Lastly, surrogating the solutions )(G of (1.56) in (7.30) and simplifying, we obtain the

following solutions:

260


,))2/tanh(()1(2

1

81632816)1(4

3

))2/tanh(()1(2

1112828)(

2

4222222

2

2

2213

C

ABABCBBCACBC

LV

CCLV

L

V

V

LBABCLVu

where

;11

0

VtLx

L, V, A, B and C are arbitrary constants.

,))2/coth(()1(2

181632816

)1(4

3

))2/coth(()1(2

1112828)(

2

4222222

2

2

2223

CABABCBBCACB

C

LV

CCLV

L

V

V

LBABCLVu

.)(sec)(tanh()1(2

1

81632816)1(4

3

)(sec)(tanh()1(2

1112828)(

2

4222222

2

2

2233

hiC

ABABCBBCACBC

LV

hiC

CLVL

V

V

LBABCLVu

Others families of exact solutions are omitted for the sake of simplicity.


,))2/tan(()1(2

1

81632816)1(4

3

))2/tan(()1(2

1112828)(

2

4222222

2

2

22123

C

ABABCBBCACBC

LV

CCLV

L

V

V

LBABCLVu

261

25.0)a( 50.0)b(

75.0)c( 0.1)d(

Fig.7.10 (a-d): Solution 123u represents the exact singular periodic traveling wave

solution for different values of and .1,1,2,1,1 VLCBA

,))2/cot(()1(2

1

81632816)1(4

3

))2/cot(()1(2

1112828)(

2

4222222

2

2

22133

C

ABABCBBCACBC

LV

CCLV

L

V

V

LBABCLVu

262

,)(sec)(tan()1(2

1

81632816)1(4

3

)(sec)(tan()1(2

1112828)(

2

4222222

2

2

22143

C

ABABCBBCACBC

LV

CCLV

L

V

V

LBABCLVu

When 0)1( C and 0 BA , the solution of (7.22) is,

,)1(

1

)1(2

81632816)1(4

3

)1(

1

)1(2112828)(

2

2

4222222

2

2

2

22253

cCC

A

ABABCBBCACBC

LV

cCC

ACLV

L

V

V

LBABCLVu


Discussion

The abundant new exact solutions for the time fractional partial differential equations

have been successfully obtained by applying the novel )/( GG -expansion method. The

nonlinear fractional complex transformation for is very important, which ensures that a

certain fractional partial differential equation can be converted into another ordinary

differential equation of integer order. The obtained solutions are more general with more

free parameters. Thus novel )/( GG -expansion method would be a trustworthy

mathematical tool for solving nonlinear evolution equations. Numerical results coupled

with graphical presentation for different values of reveals the complete reliability of

the proposed technique.

263

Chapter 8

Conclusion

264

In this thesis modified versions of )/( GG -expansion method have successfully applied

and implemented to investigate the nonlinear partial differential equations which

frequently arise in engineering sciences, mathematical physics and other scientific real

time application fields. Abundant exact traveling wave solutions including solitons,

periodic and rational solutions using symbolic computational software Maple 13 have

been constructed. The arbitrary constants in the obtained solutions imply that these

solutions have rich local structures. It has been observed that the proposed techniques are

quite useful, reliable, and efficient and can be extended for other nonlinear physical

problems. It is also observed that performance of modified versions is better than the

original scheme. Moreover, these modified algorithms give more solutions of nonlinear

partial differential equations. Computational work together with the graphical illustration

re-confirms the exactness of the proposed algorithms.

265

Chapter 9

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