conservation laws and multiple simplest equation · pdf filem. ali akbar department of ......

Nonlinear Analysis and Differential Equations, Vol. 4, 2016, no. 9, 405 - 426

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/nade.2016.6633

Conservation Laws and Multiple Simplest

Equation Methods to an Extended Quantum

Zakharov-Kuznetsov Equation

Shoukry El-Ganaini

Mathematics Department, Faculty of Science

Damanhour University, Bahira 22514, Egypt

M. Ali Akbar

Department of Applied Mathematics

University of Rajshahi, Bangladesh

Copyright © 2016 Shoukry El-Ganaini and M. Ali Akbar. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

The extended quantum Zakharov-Kuznetsov (QZK) equation, which arises in the

study of astrophysics is investigated. Exact traveling wave solutions are obtained

by using distinct types of the simplest equation method namely, the simplest

equation method, the extended simplest equation method and the modified

simplest equation method. In addition conservation laws via a new conservation

theorem are constructed for the underlying equation.

Mathematics Subject Classification: 35A20, 35D99, 35E05, 35Q40, 35Q51, 37J15

Keywords: simplest equation method, extended simplest equation method, modified

simplest equation method, conservation law, extended quantum Zakharov-Kuznetsov

(QZK) equation

1. Introduction

It is of great importance to extract exact solutions of nonlinear partial differential

equations (NPDEs) arising in mathematical physics, because many physical

phenomena are described by NPDEs. Great efforts have been developed to

different methods to solve NPDEs in the literature [1-9]. Also, Conservation laws

406 Shoukry El-Ganaini and M. Ali Akbar

have a significant role in the solution processes of NPDEs. By using the

relationship between Lie point symmetries and conservation laws, we can further

generate other conservation laws. Conservation laws describe physical conserved

quantities such as mass, energy, momentum and angular momentum as well as

charge and other constants of motion [10, 11]. A variety of powerful techniques

have been used to derive conservation laws [12-14]. The role conservation laws

have been recently used combined with double reduction theory to obtain exact

solutions of NPDEs [15-19], therefore it is very important to investigate

conservation laws for PDEs.

In this work, we consider the extended quantum Zakharov-Kuznetsov (QZK)

equation [20, 21] as

0)()( xxy

uxyy

ucyyy

uxxx

ubx

uuat

u (1)

where ba, and c are all constants, while ),,( tyxu represents the electrostatic wave

potential in plasmas. Eq.(1) supports soliton solutions that are the outcome of a

delicate balance between dispersion and nonlinearity.

The outline of this paper is as follows .In Sections 2, 3 and 4, a brief description

of the simplest equation methods for extracting traveling wave solutions of

nonlinear partial differential equations (NPDEs.) is presented. Thus, we construct

the conservation laws using a new conservation theorem in Sec.5 for the

underlying equation. Some conclusions are given in Sec.6.

2. Description of the simplest equation method [22-24]

Consider a general nonlinear PDE in the form

.0),...,,,,,,( xxx

uxt

utt

uxx

ux

ut

uuP (2)

where P is a polynomial in its arguments.

Using the traveling wave transformation 0,)(),( ctxutxu ,

where c is the wave speed ,and 0 is an arbitrary constant, to transform Eq.(2) to

a nonlinear ordinary differential equation (ODE) as

,0,...),,,( uuuuQ (3)

where the prime denotes the derivation with respect to .

Suppose the solution of Eq.(3) can be expressed as in the form

N

i

ii

u

0

))(()( , (4)

Conservation laws and multiple simplest equation methods 407

where )( satisfies either the Bernoulli or Riccati equation , N is a positive

integer that can be determined by balancing procedure and i

are parameters to

be determined , Ni ,...,2,1,0 .

Let us consider the Bernoulli equation in the following form

)(2)()( qp , (5)

where p and q are arbitrary constants. The solutions of Eq.(5) can be written as

,))

0((sinh))

0(cosh(

1

1)(

ppAq

Ap (6)

,))

0((sinh))

0(cosh(

2

)))0

((sinh))0

(cosh(()(

ppAq

ppp (7)

where 1

A ,2

A and 0 are arbitrary constants.

Also let us consider the Riccati equation as

,)())(2)( sqp (8)

where qp , and s are constants. It should be mentioned that Eq.(8) has the

following solutions [25]

,)]0

()2/tanh[()2/()2/()( ppq (9)

,))2/((sinh)/2())2/((cosh

))2/((sec

)])2/tanh[()2/()2/()(

pC

h

ppq

(10)

where spq 422 .

Substituting (4) into (3) with (5) (or (8)), then the left-hand side of (3) is reduced

to a polynomial in )( . Setting each coefficient of this polynomial to zero,

yields a system of algebraic equations which can be solved for qpi

,, . Hence,

substituting these values into (4), the exact traveling wave solutions are reported

then for Eq.(2).

Note. If we put Ap and 1q in Bernoulli equation (5), we obtain

)(2)()( A , (11)

which has the exact solution


,))]0

()2/tanh((1[)2/()( AA (12)

provided 0A , and the other exact solution

,))]0

()2/tanh((1[)2/()( AA (13)

if 0A .

Application of the simplest equation method using Bernoulli equation

Introducing a wave variable

0 vtrykx , to convert (1) an ordinary differential equation(ODE)

0))()33(( urkrkcrkbuakuuv (14)

Integrating (2) with respect to and setting the integration constant to zero, yields

0))()33((2)2

1( urkrkcrkbuakvu . (15)

Thus, we are concerned to solve the equation (15).

Considering the homogeneous balance between u and 2u , we get 2N ,

therefore , the solution of (24) can be written as

)(22

)(10

)( u (16)

Substituting (16) into (15) and making use of the Bernoulli equation (5) and then

equating the coefficients of the functions )( i to zero, we obtain the following

system of algebraic equations

)21(0202

1

0:)(0

)20(01

2101

:)(1

)19(0)1

32

24(2

)20

221

(2

1:)(2

)18(0)2

101

22(21

:)(3

)17(02

26222

1:)(4

kav

Mpkav

Mqppvka

Mqpqka

Mqka

.))()33(( rkrkcrkbM (22)


On solving the above system of over-determined algebraic equations, we obtain

two sets of solution,

,,)/12(,)/12(,)/2( 2221

20 MpvMkaqMkaqpMkap

(23)

.,)/12(,)/12(,0 22210 MpvMkaqMkaqp (24)

Therefore, using solutions (6) and (7) of (5) and the values of 210 ,, from

(23) into solution ansatz (16), we obtain the exact solutions of (1) in the form:

,

2

))0

((sinh))0

(cosh(1

1)/212(

))0

((sinh))0

(cosh(1

1)/12(

)/22(),,(1

ppAq

ApMkaq

ppAq

ApMkaqp

Mkaptyxu

(25)

2

))0

((sinh))0

(cosh(2

)))0

((sinh))0

(cosh(()/212(

))0

((sinh))0

(cosh(2

)))0

((sinh))0

(cosh(()/12(

)/22(),,(2

ppAq

pppMkaq

ppAq

pppMkaqp

Mkaptyxu

(26)

where .))()33((,0

)2( rkrkcrkbMtMpyrxk

In the same way, using solutions (6) and (7) of (5) and the values of 210 ,, ααα

from (24) into solution ansatz (16), we obtain the following exact solutions of (1):

,

2

))0

((sinh))0

(cosh(1

1)/212(

))0

((sinh))0

(cosh(1

1)/12(),,(3

ppAq

ApMkaq

ppAq

ApMkaqptyxu

(27)


2

))0

((sinh))0

(cosh(2

)))0

((sinh))0

(cosh(()/212(

,))

0((sinh))

0(cosh(

2

)))0

((sinh))0

(cosh(()/12(),,(

4

ppAq

pppMkaq

ppAq

pppMkaqptyxu

(28)

where .))()33((,)( 2 rkrkcrkbMtMpyrxk

and 1,,,,,,, Akrqpcba and 2A are arbitrary constants.

Application of the simplest equation method using Riccati equation

In the case of Riccati equation, the solution ansatz can be written as,

)(22

)(10

)( u (29)

Substituting (29) into (15) and making use of the Riccati equation (8) and then

equating the coefficients of the functions )( i to zero, we thus obtain a system

of algebraic equations and by solving it, we obtain

)30(,2/2

,)(212

2)33(212

,23,1

)/(2

,11

,00

sqp

rkprk

karkpbcMqvqp

where .))()33(( rkrkcrkbM

Thus, using solutions (9) and (10) of (8), (30) and (16)), yields following exact

solutions of Eq. (1):

)32(

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

))2/((sec

)])2/tanh[()2/()2/(()/(

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

))2/((sec

)])2/tanh[()2/()2/

1),,(

6

)31(2)])

0()2/tanh[()2/()2/(()/(

)]0

()2/tanh[()2/()2/(

1),,(

5

pC

h

ppqqp

pC

h

ppq

tyxu

ppqqp

ppq

tyxu


where

spqrkrkcrkbMtMqyrxk 42.,))()33((,0

)23( , c

and p are determined in (39) , Cksrqpba ,1

,,,,,,, are arbitrary constants.

3 The modified simplest equation method [26,27]

Step1.We suppose that (15) admits the formal solution

,

0

)(

iN

ii

a

(33)

where i

a are constants to be determined such that 0N

a Ni ,...,1,0, .

The function )( is an unknown function to be determined later provided that

0)( .

Step 2. We substitute (33) into (15), we calculate all the necessary derivatives

,...,, and then we determine the function )( .

As a result, we get a polynomial of ,...)1,0()(

jj , with the derivatives of

)( .

Setting all the coefficients of ,...)1,0()(

jj , to zero yields a system of

algebraic equations which can be solved to obtain i

a and )( .

Step3. Substituting the values of i

a and )( and its derivative )( into (33)

leads to exact traveling wave solutions of (1).

Application of the modified simplest equation method

Suppose that (15) admits the formal solution

,

2

)(

)(

2)(

)(

10)(

aaau (34)

where .10

aa and 2

a are constants to be determined such that 02a .

Substituting (34) into (15) and equating all the coefficients of

,4

,3

,2

,1

,0 to zero, we respectively obtain

,0202

1

0:0 akaav (35)


,01101

:1

aMaakaav (36)

,022

22

2

1322

12

1220

22

:2

aMMa

aMakaaakaav (37)

022

1031

2321

:3

aMMaaaka (38)

042

64222

1:

4

Makaa (39)

where .))()33(( rkrkcrkbM

From (35) and (39), we deduce that

kaMakavaa /122

,/20

,00

, (40)

provided 0ka . Now we consider the following cases:

Case 1. When 00a , 0

1a , and 0 , then we deduce from Eqs.(36-38) that

,0)( 0 Makav (41)

0)2()/224(1

32)212

1/12( kaMMaakakaMv (42)

0121

Maka (43)

From (41) and (43) we have

10

12

aka

M

akav

M

(44)

and consequently we get

112 akav (45)

Integrating (45), we get

1

12

1

aka

v

eC , (46)

and substituting from (46) into (44), we get


)

1

12(

11

12)(

aka

v

eCaka

M (47)

Integrating (47), we obtain

,

)

1

12(

12)(

aka

v

eCv

MC (48)

where 1C and 2C are arbitrary constants of integration.

Substituting (44) into (42), we have

,)/12(1 Mvkaa (49)

provided 0ka .

Thus the exact solution of (1) in this case has the form

2

)()

1

12(

)/1

(2

)()

1

12(

)21

33/321

1728(

)()

1

12(

)/1

(2

)()

1

12(

)/1

12()(

0

0

0

0

tvyrxkaka

v

evMCC

tvyrxkaka

v

eakaMC

tvyrxkaka

v

evMCC

tvyrxkaka

v

ekaMCu

(50)

where the value of 1a is provided in (49).

Since 1C and 2C are constants of integration, we might arbitrarily choose their

values. Therefore, if we choose ,/1

,12

MvCC we have respectively the

solitary wave solution

2

))(

1

6(tanh1)2

133/2432(

))(

1

6(tanh1)/6()(

1

0

0

tvyrxkaka

vakaMv

tvyrxkaka

vkavu

(51)


2

))(

1

6(coth1)2

133/2432(

))(

1

6(coth1)/6()(

2

0

0

tvyrxkaka

vakaMv

tvyrxkaka

vkavu

(52)

Case 2. When 00a , 0

1a , and 0 , then we deduce from Eqs.(36) to (38)

that

,0 Mv (53)

0)2(2241

32)212

112( 22 MkaMaakaMv (54)

0121

Maka (55)

Consequently, we deduce from Eqs. (53) and (55) that

(56)

and thus, we get

112 akav (57)

Integrating (57), we then have

)

1

12(

1

aka

v

eC

. (58)

From (58) into (56), we get

)

1

12(

11

12)(

aka

v

eCaka

M

(59)

Integrating (59), we obtain

,

)

1

12(

12)(

aka

v

eCv

MC

(60)

where 1C and 2C are an arbitrary constants of integration.

Consequently, we conclude from (54) that

1

12

aka

M

v

M


1,3)/12(1

iMvkaia (61)

provided 0ka .

Thus, the exact solution of (1) in this case has the form

2

)()

1

12(

)/1

(2

)()

1

12(

)21

33/321

1728(

)()

1

12(

)/1

(2

)()

1

12(

)/1

12()/2()(

0

0

0

0

tvyrxkaka

v

evMCC

tvyrxkaka

v

eakaMC

tvyrxkaka

v

evMCC

tvyrxkaka

v

ekaMCkavu

(62)

where 1

a is as given in (61).

The value of 1C and 2C can be selected randomly as they are integration

constant. Therefore, if we select ,/1

,12

MvCC we have the solitary wave

solution

)63(,2

))(

1

6(coth1)2

122/216(

))(

1

6(coth34

)/2()(1

0

0

tvyrxkaka

vakavM

tvyrxkaka

v

kavu

4. The extended simplest equation method [28, 29]

Suppose the solution of Eq.(5) can be expressed as another

ansatz in the following form

)64(2

))(

1

6(tanh1)2

122/216(

))(

1

6(tanh34

)/2()(2

0

0

tvyrxkaka

vakavM

tvyrxkaka

v

kavu


,11

00

)(

jN

jj

iN

ii

u (65)

where i

, j

are constants to be determined later, and

01

NN , 1,...,2,1,0,,...,2,1,0 NjNi .

The function )( satisfies the second order linear ODE

, (66)

where and are constants.

Eq.(66) has the following three types of general solutions with double arbitrary

parameters, namely,

0,21

)2/(

0,/)(sin2

)(cos1

0,/)(sinh2

)(cosh1

)(

2

AA

AA

AA

, (67)

0,/2

21

)22

221

(

0,/2

21

)/222

21

(

0,/2

21

)/222

21

(

2

AA

AA

AA

(68)

where 1

A and 2

A are arbitrary constants.

By substituting (65) into (3) and using the second order linear ODE (66) and

Eq.(68 ), the left-hand side of (3) is transformed to a polynomial in i/1 and

)/()/1( i . Setting each coefficient of this polynomial to zero , yields a set of

algebraic equations which can be solved for the constants

,0

,)1,...,2,1,0,,...,2,1,0(, vNjNiji

and .Therefore, substituting the

resultant values of these constants and the general solutions (67) of (66) into (5),

we can obtain the exact traveling wave solutions of Eq.(4) in a concise manner.


Application of the extended simplest equation method Suppose the solution of (3) is of the form

,1

4

1

3

2

210)(

u (69)

where 0

, 1

, 2

, 3

, and 4

are constants to be determined later and the

function )( satisfies the linear ODE (66).

By substituting (69) into (3) and using (66)) and (68), the left-hand side of (3)

become a polynomial in i/1 and )/()/1( i . Putting every coefficient of this

polynomial as zero, yields a system of algebraic equations where by solving it via

Mathematica 9, we therefore have the following cases:

If 0 , we obtain

12

,12

,04

,03

,00

AAi

, , 0,

2 akav , (70)

δ

AδAi

2

129

2,

223

3,0

4,0

1,0

0

2

,

rk

rkrkbc

)22( ,

22

1kav , .

23

i (71)

Substituting (70) and (71) into (69) and making use of solutions (67) of (66), then

the exact traveling wave solutions which expressed by hyperbolic functions are

obtained and can be written as

(72),0

)1

(

,)(sinh)(cosh

)(cosh)(sinh1

,)(sinh)(cosh

)(cosh)(sinh

1)(

1

tkaiyrxk

u

)73(0

)2

1(

,

2/3)(sinh2

129

)(cosh1

23

2

2/3)(sinh2

129

)(cosh1

)(cosh2

129

)(sinh1

2)(

2

2

2

2

2

tkayrxk

iδ

AδA

i

iδ

AδA

δ

AδA

u


where 2

,12

,1

AA , and 0 are an arbitrary constants.

If 0 , we obtain

12

,12

,04

,03

,00

AAi

, , 0,

2 kav , (74)

δ

AδAi

2

129

2,

223

3,0

4,0

1,0

0

2

,

rk

rkrkbc

)22( ,

22

1kav , .

23

i (75)

Substituting (74) and (75) into (69) and making use of solutions (67) of (66), the

exact traveling wave solutions expressed by trigonometric functions are obtained

as

(76),0

)1

(

,)(sin)(cos

)(cos)(sin1

,)(sin)(cos

)(cos)(sin

1)(

1

tkaiyrxk

i

iiu

where 2

,12

,1


If 0 , we obtain

1

3

22

,04

,01

,00

AiA

, 2

22/2

1AA (78)

Substituting (78) into (69) and making use of solutions (67) of (66), we obtain the

rational function solutions of Eq.(1) and can be written as

)77(0

)2

1(

,

2/3)(sin2

129

)(cos1

23

2

2/3)(sin2

129

)(cos1

)(cos2

129

)(sin1

2)(

2

2

2

2

2

tkayrxk

iδ

AδA

i

iδ

AδA

δ

AδA

u


,2

23/

242

242

1

)1

/22

(3

42

223

/2

422

421

22

21

24)(

AiAA

AA

AiAA

AAu

.0 tvyrxk (79)

where 2

,13

,2


5. Conservation laws for the extended quantum Zakharov-

Kuznetsov (qZK) equation

Preliminaries[30]

Consider an thr -order partial differential equation

0))(

,...,)1(

,,( r

uuuxF (80)

with independent variables ),...,1( nxxx and one dependent variable u , where

},...{)2(

},{)1( ij

uui

uu denote the set of partial derivatives of the first order,

second order ,etc. , and ./2,/j

xixuij

uixui

u

The adjoint equation to (80) is

0))(

,)(

,...,)1(

,)1(

,,,( r

vr

uvuvuxF (81)

where

uvFr

ur

uvuvuxF )())(

,)(

,...,)1(

,)1(

,,,(* (82)

and

1 ...1

...

1

)1(

rrii

uri

Di

Dr

uu

(83)

denote the Euler-Lagrange operator, and v is a new dependent variable. Here

...

juij

uui

uix

iD (84)

are the total differentiations with respect to niix ,...,1, .

Eq.(80) is called self-adjoint if the equation

0))(

,)(

,...,)1(

,)1(

,,,(* r

ur

uuuuuxF (85)


obtained from the adjoint equation (82) by the substitution uv ,

is identical to the original equation (81). This concept can be written in an

equivalent form as

))(

,...,)1(

,,(,...))1(

,,())(

,)(

,...,)1(

,)1(

,,,(*r

uuuxFuuxr

ur

uuuuuxF (86)

Theorem of Ibragimov on conservation laws [31]:

Every Lie point, Lie –Backlund or non-local symmetry

uuux

ixuuxiX

,...)

)1(,,(,...)

)1(,,( (87)

of Eq.(80) provides a conservation law 0)( iCi

D , for the system of equations

(81) and (82).

The conserved vector is given by

(88)......)(

...)()(...)()(

ijku

LW

kD

jD

ijku

L

kD

iju

LW

jD

ijku

L

kD

jD

iju

L

jD

iu

LWLiiT

where W (the characteristic function) and L (the Lagrangian) are defined as

.,...,1),)(

,...,)1(

,,(, njr

uuuxFvLj

uj

W (89)

Let us recall the symmetry algebra of Eq.(1) as given in [29] which is spanned by

the five vector fields

(90))./)(/1())/(5

),/(2)/()/()/(34

,/3

,/1

,/1

uaxtX

uuyyxxttXyXxXtX

Now, the extended quantum Zakharov-Kuznetsov equation (1) together with its

adjoint equation are given by

0)()( xxy

uxyy

ucyyy

uxxx

ubx

uuat

uF (91)

0)()(* xxy

vxyy

vcyyy

vxxx

vbx

vuat

vF (92)

The third-order Lagrangian L for the system of equations (81) and (82) is given

by

))()((xxy

uxyy

ucyyy

uxxx

ubx

uuat

uvFvL (93)


Therefore, we must consider the following five cases:

(i) Considering the vector field tX /1

, for Eq.(1) .By using (88) and taking

uv ,due to self-adjointness of Eq.(1), the resultant conserved vector is given by

xxycuu

xyycuu

yyybuu

xxxuub

xuuatC 2

1,

yytcuu

xytcuu

xxtbuu

ytu

yuc

ytu

xuc

xtu

ycu

xtu

xbu

yyu

tcu

xyu

tcu

xxu

tbu

tuuaxC

2

221

,

yytubu

xytucu

xxtucu

ytu

yub

ytu

xuc

xtu

yuc

xtu

xuc

yyu

tbu

xyu

tcu

xxu

tcuyC

2

21

(94)

(ii) Using the Lie point symmetry generator xX /2

, for Eq.(1) and using

(88) , the components of the conserved vector are given by

xuutC

2

xyu

ycu

yu

xxuc

yyu

xcu

xyu

xcu

xxycuu

yyybuu

tuuxC

2

xyybuu

xxycuu

xxxcuu

xyu

ybu

yyu

xbu

xyu

xcu

xxu

ycuyC 2

2

(95)

(iii) Using yX /3

, for Eq.(1) and (88) , the component of the conserved

vector are obtained and can be written as

yuutC

3,

yyycuu

xyycuu

xxybuu

yyu

xcu

xyu

xbu

xyu

ycu

xxu

ybu

yuuaxC 22

3

yyu

xcu

xyu

xcu

xyu

ycu

xxu

ycu

xyycuu

xxxbuu

xuau

tuuyC 2

3

(96)

(iv) The Lie point symmetry generator

)/(2)/()/()/(34

uuyyxxttX

For Eq.(1), leads to a conserved vector with component given by


yuyu

xuxuu

xxyctuu

xyyctuu

yyybtuu

xxxbtuu

xuuattC 22333323

4

.326

33333

262336

3612

63223234

yyycyuu

yytctuu

xyycyuu

xxyxcuu

xytctuu

xxyybuu

xxtbtuu

yyyxbuu

ytu

yctu

ytu

xctu

xtu

yctu

xtu

xbtu

yuayu

yu

xcu

tuatu

yyu

tctu

yyu

xcyu

yyu

xcxu

yycuu

xxu

ycxu

xxu

yybu

xxu

tbtu

xxbuu

xycuu

xyu

xbyu

xyu

ycxu

xyu

tctu

xyu

ycyu

xyu

xcxuau

ycu

xbu

tuxuxC

(97).3

263

33363

36612

36232324

xyyxbuu

yytbtuu

xxycxuu

xytctuu

xxxcxuu

xxtctuu

xyycyuu

xxxybuu

ytu

ybtu

ytu

xctu

xtu

yctu

yu

xcu

xtu

xctu

yyu

xcyu

yyu

xxbu

yyu

tbtu

yybuu

xyu

yxbu

xyu

xcyu

xyu

tctu

xyu

ycyu

xyu

xcxu

xycuu

xxu

ycxu

xxu

ycyu

xxu

tctu

xxcuu

ybu

xcu

xuuay

tuyuyC

(v) The last symmetry generator )/)(/1())/(5

uaxtX , give the

following conseved quantities

xutuatC )/1(

5,

,

)/()/2()/(25

xxyctuu

yyybtuu

xyu

yctu

xxu

yctu

yyu

xctu

xyu

xctu

yyuac

xyuac

xxuab

ttuuuxC

)98(.2

)/()/2()/(5

xyybtuu

xxyctuu

xxxctuu

xyu

ybtu

xxu

yctu

yyu

xbtu

xyu

xctu

yyuab

xyuac

xxuacyC

6. Conclusion

In this article, we have obtained wider classes of exact traveling wave solutions of

the extended qZK equation by using distinct types of the simplest equation

methods. In addition, we have constructed conservation laws for the extended

qZK equation via the new conservation theorem. The conservation laws can be

associated with Lie symmetry generators of NPDEs to yield another conservation

laws and more exact solutions of these nonlinear equations with the aid of double

reduction theory and Kara and Mahomed results. These useful ideas are in urgent

need to be conducted in a separate study in a future work.


Competing interests Authors have declared that no competing interests exist.

Authors’ contributions This research work was carried out in collaboration between the authors. Both

authors have a good contribution to design the study and to perform the analysis

of this research work. Both the authors read and approved the final manuscript.

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Received: June 20, 2016; Published: July 25, 2016

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