World Applied Sciences Journal 22 (3): 396-407, 2013ISSN 1818-4952© IDOSI Publications, 2013DOI: 10.5829/idosi.wasj.2013.22.03.10413

Corresponding Author: Sayeda Sultana, Department of Economics, Kalachandpur School and College, Dhaka, Bangladesh.Tel: +88 01718817903.

396

Traveling Wave Solutions for the Nonlinear Evolution Equationvia the Generalized Riccati Equation and the (G'/G)-Expansion Method

M.M. Zaman and Sayeda Sultana1 2

Fortunex Limited, Baridhara, Bangladesh1

Department of Economics, Kalachandpur School and College, Dhaka, Bangladesh2

Abstract: In this article, we investigate the simplified Modified Camassa-Holm (MCH) equation involvingparameters by applying the (G'/G)-expansion method together with the generalized Riccati equation. In addition,G'( ) = c + aG( ) + bG ( ) is used, as an auxiliary equation and called generalized Riccati equation, where a,2

b and c are arbitrary constants. The obtained traveling wave solutions including solitons and periodic solutionsare presented through the hyperbolic, the trigonometric and the rational functions. Further, it is important topoint out that one of our solutions is in good harmony for a special case with already published results whichin turn validates our other solutions. Moreover, the numerical presentations have been demonstrated of someof the solutions with the aid of commercial software Maple.

Mathematics Subject Classification: 35K99 35P99 35P05PACS: 02.30.Jr 05.45.Yv 02.30.IkKey words: The generalized Riccati equation The simplified MCH equation The (G'/G)-expansion method

Traveling wave solutions Nonlinear partial differential equations

INTRODUCTION G''( )+ G'( ) + µG(µ) = 0, as an auxiliary equation. Later

A great deal of work has been carried out in order to method for obtaining exact traveling wave solutions ofdetermine exact traveling wave solutions for nonlinear many nonlinear PDEs. For example, Zayed and Al-Joudipartial differential equations (PDEs) in all areas of science [33] applied this method to find traveling wave solutionsand engineering, such as, plasma physics, chemical for some nonlinear partial differential equations inphysics, optical fibres, solid state physics, fluid mathematical physics whilst Zayed and Gepreel [34]mechanics, chemistry and many others [1-52]. In the implemented the same method to obtain wave solutionsrecent past, many researchers implemented various for the NLEEs. Naher et al. [35] investigated themethods to establish traveling wave solutions of different Caudrey-Dodd-Gibbon equation by using this methodnonlinear PDEs. For example, the Jacobi elliptic function for constructing abundant traveling wave solutions.expansion method [1, 2], the Hirota’s bilinear Feng et al. [36] concerned about the same method for thetransformation method [3], the inverse scattering method Kolmogorov-Petrovskii-Piskunov equation to establish[4], the weierstrass elliptic function method [5], the analytical solutions. Ozis and Aslan [37] studied thetanh-coth method [6], the variational iteration method Kawahara type equation by using this method to obtain[7-15], the direct algebraic method [16], the Cole-Hopf exact solutions. In Ref. [38], Zhang et al. implemented thetransformation method [17], the Exp-function method improved (G'/G)-expansion method to construct[18-22] and others [23-31]. traveling wave solutions of nonlinear evolution

Recently, Wang et al. [32] introduced the equations. Naher et al. [39] constructed analytical(G'/G)-expansion method to construct traveling wave solutions of the higher dimensional NLEE by using thesolutions for the nonlinear evolution equations (NLEEs). same method. In Ref. [40], Naher and AbdullahIn this method, they employed second order linear investigated the nonlinear reaction diffusion equation toordinary differential equation with constant coefficients obtain traveling wave solutions via this method while

on, many researchers used the useful (G'/G)-expansion

( ), , , , , ,... 0,t x xt tt xxS u u u u u u =

( ) ( ), , ,u x t q x W t= = −

( ), ', , ,... 0,X q q q q′′ ′′′ =

( )0

' ,in

ii

Gq fG

=

= ∑

2,G c aG bG′ = + +

World Appl. Sci. J., 22 (3): 396-407, 2013

397

Naher and Abdullah [41] executed the same Step 1: Consider the traveling wave variable:method to study the (2+1)-dimensional modifiedZakharov-Kuznetsov equation for obtaining exact (2)solutions and so on.

Zhu [42] presented the generalized Riccati equation where W is the speed of the traveling wave. Now, usingmethod to construct non- traveling wave solutions for the Eq. (2), the Eq. (1) is transformed into an ordinary(2+1)-dimensional Boiti-Leon-Pempinelle equation. In the differential equation for q( )method, G'( ) = c + aG( ) + bG ( ) is utilized, as the2

auxiliary equation, where a, b and c are arbitrary (3)constants. Salas [43] applied the projective Riccatiequation to obtain exact solutions for a type of where the superscripts stand for the ordinary derivativesgeneralized Sawada-Kotera equation while Gomez et al. with respect to [44] studied the higher-order nonlinear KdV equation toconstruct traveling wave solutions by using the method. Step 2: According to possibility, Eq. (3) integrates termIn Ref. [45], Li and Dai implemented the generalized Riccati by term one or more times, yields constant(s) ofequation mapping with the (G'/G)-expansion method to integration. The integral constant(s) may be zero, forconstruct traveling wave solutions for the higher simplicity.dimensional Jimbo-Miwa equation whilst Guo et al. [46]investigated the diffusion-reaction and mKdV equations Step 3: Suppose that the traveling wave solution ofwith variable coefficient by using the extended Riccati Eq. (3) can be expressed in the following form:equation mapping method. Naher and Abdullah [47]implemented the extended generalized Riccati equationmapping method to construct traveling wave solutions of (4)the modified Benjamin-Bona-Mahony equation and so on.

Many researchers investigated the simplified MCHequation by using different methods to establish exact where f (i = 0,1,2,...,n) and f 0, with G = G( ) is thesolutions. For example, Liu et al. [48] concerned about the solution of the generalized Riccati equation:(G'/G)-expansion method to solve the simplified MCHequation, whereas, the second order linear ordinary (5)differential equation (LODE) is considered as an auxiliaryequation. In Ref. [49], Wazwaz studied this equation by where a, b, c are arbitrary constants and b 0.using the sine-cosine algorithm.

The significance of our present paper is, in order to Step 4: To determine the positive integer n, considerconstruct new results, a simplified MCH equation is the homogeneous balance between the nonlinearconsidered by applying the (G'/G)-expansion method terms and the highest order derivatives appearing intogether with the generalized Riccati equation. Moreover, Eq. (3).we have constructed new exact traveling wave solutionsincluding solitons, periodic and rational solutions. Step 5: Substituting Eq. (4) along with Eq. (5) into the

The (G'/G)-expansion Method Together With the order, the left hand side of Eq. (3) converts intoGeneralized Riccati Equation: Suppose the general polynomials in G'( ) and G ( ), (l = 0,1,2,...).nonlinear partial differential equation: Then equating each coefficient of the polynomials to

(1) a,b,c and W.

where u = u(x,t) is an unknown function, S is a polynomial Step 6: Solve the system of algebraic equations which arein u = u(x,t) and the subscripts signify the partial found in Step 5 with the aid of algebraic softwarederivatives. Maple to obtain values for f (i = 0,1,2,...,n) and W.

The main steps of the (G'/G)-expansion method Then, substitute obtained values in Eq. (4) along with Eq.together with the generalized Riccati equation mapping (5) with the value of n, we can obtain exact solutions of[32, 42] are as follows: Eq. (1).

i n

Eq. (3), then collect all the coefficients with the same

1

zero, yield a set of algebraic equations for f (i = 0,1,2,...,n),i

i

22

11 44 tanh .

2 2a bcG a a bc

b

− − = + −

22

21 44 coth .

2 2a bcG a a bc

b

− − = + −

( ) ( )2 2 23

1 4 tanh 4 sec 4 .2

G a a bc a bc i h a bcb− = + − − ± −

( ) ( )2 2 24

1 4 coth 4 csc 4 .2

G a a bc a bc h a bcb− = + − − ± −

2 22

51 4 42 4 tanh cot .

4 4 4a bc a bcG a a bc h

b

− − − = + − +

( )( ) ( )( )

2 2 2 2 2

6 2

4 4 cosh 41 ,2 sinh 4

P Q a bc P a bc a bcG a

b P a bc Q

± + − − − − = − +

− +

( )( ) ( )( )

2 2 2 2 2

7 2

4 4 cosh 41 ,2 sinh 4

P Q a bc P a bc a bcG a

b P a bc Q

± + − + − − = − −

− +

2

8 2 22

42 cosh2

.4 44 sinh cosh

2 2

a bcc

Ga bc a bca bc a

− =

− − − −

2

9 2 22

42 sinh2

.4 4sinh 4 cosh

2 2

a bcc

Ga bc a bca a bc

− − =

− − − −

( )( ) ( )

2

10 2 2 2 2

2 cosh 4.

4 sinh 4 cosh 4 4

c a bcG

a bc a bc a a bc i a bc

−=

− − − − ± −

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398

We have the following twenty seven solutions including four different families of Eq. (5).

Family 2.1: When a – 4bc>0 and ab 0 or bc 0, the solutions of Eq. (5) are:2

where P and Q are two non-zero real constants.

( )( ) ( )

2

11 2 2 2 2

2 sinh 4.

sinh 4 4 cosh 4 4

c a bcG

a a bc a bc a bc a bc

−=− − + − − ± −

2 2

12 2 2 22 2 2

4 44 sinh cosh4 4

.4 4 42 sinh cosh 2 4 cosh 4

4 4 4

a bc a bcc

Ga bc a bc a bca a bc a bc

− − =

− − − − + − − −

22

131 44 tan .2 2

bc aG a bc ab

− = − + −

2

214

1 44 cot .2 2

bc aG a bc ab

− − = + −

( ) ( )2 2 215

1 4 tan 4 sec 4 .2

G a bc a bc a bc ab = − + − − ± −

( ) ( )2 2 216

1 4 cot 4 csc 4 .2

G a bc a bc a bc ab− = + − − ± −

2 22

171 4 42 4 tan cot .4 4 4

bc a bc aG a bc ab

− − = − + − −

( )( ) ( )( )

2 2 2 2 2

18 2

4 4 cos 41 ,2 sin 4

P Q bc a P bc a bc aG a

b P bc a Q

± − − − − − = − +

− +

( )( ) ( )( )

2 2 2 2 2

19 2

4 4 cos 41 ,2 sin 4

P Q bc a P bc a bc aG a

b P bc a Q

± − − + − − = − −

− +

2

20 2 22

42 cos2

.4 44 sin cos

2 2

bc ac

Gbc a bc abc a a

− − =

− − − +

2

21 2 22

42 sin2

.4 4sin 4 cos

2 2

bc ac

Gbc a bc aa bc a

− =

− − − + −

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399

Family 2.2: When a – 4bc < 0 and ab 0 or bc 0, the solutions of Eq. (5) are:2

where P and Q are two non-zero real constants and satisfies P – Q >0.2 2

( )( ) ( )

2

22 2 2 2 2

2 cos 4.

4 sin 4 cos 4 4

c bc aG

bc a bc a a bc a bc a

− −=

− − + − ± −

( )( ) ( )

2

23 2 2 2 2

2 sin 4.

sin 4 4 cos 4 4

c bc aG

a bc a bc a bc a bc a

−=− − + − − ± −

2 2

24 2 2 22 2 2

4 44 sin cos4 4

.4 4 42 sin cos 2 4 cos 4

4 4 4

bc a bc ac

Gbc a bc a bc aa bc a bc a

− − =

− − − − + − − −

( ) ( )( )25 ,cosh sinh

apGb p a a

−=

+ −

( ) ( )( )( ) ( )( )26

cosh sinh,

cosh sinha a a

Gb p a a− +

=+ +

271

1 ,Gb g

−=

+

22 0, where , 0.t x x x t xu k u u u u k+ − + = ∈ >

22 0.W q k q Wq q q′ ′ ′′′ ′− + + + =

( ) 32 0,3

k W q W q q C′′− + + + =

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400

Family 2.3: when c = 0 and ab 0, the solutions of Eq. (5) become:

where p is an arbitrary constant.

Family 2.4: when b 0 and c = a = 0, the solution of Eq. (5) becomes:

where g is an arbitrary constant.1

Application of the Method: In this section, we investigate the simplified MCH equation by using the method to constructnew exact traveling wave solutions.

The Simplified MCH Equation: In the present work, we consider the simplified MCH equation with parameters followedby Liu et al. [48]:

(6)

Details of CH and MCH equations can be found in references [48-52].Now, we use the transformation Eq. (2) into the Eq. (6), which yields:

(7)

Eq. (7) is integrable, therefore, integrating with respect once yields:

(8)

where C is an integral constant which is to be determined later.Taking the homogeneous balance between q'' and q in Eq. (8), we obtain n = 1.3

( ) ( )1 0 1'/ , 0.q f G G f f= + ≠

( ) ( )11 0,q f a cG bG f−= + + +

0 12 2 2 26 6 4 6, 2 , , 16 ,

2 8 2 8 2 8 2 8k k k kf ai f i W C iabc

a bc a bc a bc a bc= = ± = = ±

+ + + + + + + +

( )( )

2 2

1 1 02 sec

,2 tanh

hq f f

aΦ Φ

= ++ Φ Φ

1 2 42

a bcΦ = −6 ,0 2 2 8

kf aia bc

=+ +

621 2 2 8

kf ia bc

= ±+ +

42 2 8

kx ta bc

= −

+ +

( )( )

2 2

2 1 02 csc

.2 coth

hq f f

a− Φ Φ

= ++ Φ Φ

( ) ( )( )( ) ( )

2

3 1 04 sec 2 1 sin 2

.cosh 2 2 sinh 2 2

h i hq f f

a iΦ Φ Φ

= +Φ + Φ Φ ± Φ

( )( ) ( )

2

4 1 02 csc

.sinh 2 cosh

hq f f

a− Φ Φ

= +Φ + Φ Φ

( )( )

2

5 1 04 csc 2

.tanh 2

hq f f

a− Φ Φ

= +Φ + Φ

( ) ( ) ( )

( )( ) ( ) ( ) ( )

2 2 2

6 1 02 2

4 sinh 2 cosh 2,

sinh 2 sinh 2 2 2 cosh 2

P P Q P Qq f f

P Q aP aQ P Q P

− Φ − Φ − + Φ = + Φ + Φ + − Φ + + Φ Φ

( ) ( ) ( )

( )( ) ( ) ( ) ( )

2 2 2

7 1 02 2

4 sinh 2 cosh 2,

sinh 2 sinh 2 2 2 cosh 2

P P Q P Qq f f

P Q aP aQ P Q P

− Φ − Φ + + Φ = + Φ + Φ + + Φ + + Φ Φ

World Appl. Sci. J., 22 (3): 396-407, 2013

401

Therefore, the solution of Eq. (8) is of the form:

(9)

Using Eq. (5), Eq. (9) can be re-written as:

(10)

where a,b and c are free parameters.By substituting Eq. (10) into Eq. (8), collecting all coefficients of G and G (l = 0,1,2,...) and setting them equal tol 1

zero, we obtain a set of algebraic equations for f ,f ,a,b,c,C and W (algebraic equations are not presented, for simplicity,).0 1

With the aid of algebraic software Maple, solving the system of algebraic equations, we obtain

where > 0 and i is an imaginary unit.

Family 3.1: The soliton and soliton-like solutions of Eq. (6) (when a – 4bc > 0 and ab 0 or bc 0) are:2

where , , and a,b,c are arbitrary constants.

( )( ) ( )

2

8 1 02 sec

.2 sinh cosh

hq f f

a− Φ Φ

= +Φ Φ − Φ

( )( ) ( )

2

9 1 02 csc

.2 cosh sinh

hq f f

aΦ Φ

= +Φ Φ − Φ

( ) ( )( )( ) ( )

2

10 1 04 sec 2 1 sinh 2

.cosh 2 2 sinh 2 2

h iq f f

a iΦ Φ Φ

= +Φ − Φ Φ Φ

( ) ( )( )( ) ( )

2

11 1 04 csc 2 1 cosh 2

.2 cosh 2 sinh 2 2

hq f f

aΦ Φ Φ

= +Φ Φ − Φ ± Φ

( ) ( )( )

2

12 1 02 csc sec

.2 tanh

h hq f f

aΦ Φ Φ

= +Φ− Φ

( )( )

2 2

13 1 02 sec

,2 tan

q f fa− ∆ ∆

= +− + ∆ ∆

1 24 ,2

bc a∆ = −6 ,0 2 2 8

kf aia bc

=+ +

621 2 2 8

kf ia bc

= ±+ +

42 2 8

kx ta bc

= −

+ +

( )( )

2 2

14 1 02 csc

,2 cot

q f fa∆ ∆

= ++ ∆ ∆

( ) ( )( )( ) ( )

2

15 1 04 sec 2 1 sin 2

.cos 2 2 sin 2 2

q f fa∆ ∆ ± ∆

= +− ∆ + ∆ ∆ ± ∆

( )( ) ( )

2

16 1 02 sec

,cos 2 sin

q f fa

− ∆ ∆= +

∆ + ∆ ∆

( )( ) ( )

2

17 1 02 csc

,sin 2 cos

q f fa

− ∆ ∆= +

∆ + ∆ ∆

( ) ( )( )( )( ) ( ) ( )( )

2 2 2

18 1 02 2

4 cos 2 sin 2,

sin 2 sin 2 2 cos 2 2

P P Q Q Pq f f

P Q aP P aQ P Q

∆ − ∆ − ∆ −= +

∆ + ∆ + ∆ ∆ + − ∆ −

( ) ( )( )( )( ) ( ) ( )( )

2 2 2

19 1 02 2

4 cos 2 sin 2,

sin 2 sin 2 2 cos 2 2

P P Q Q Pq f f

P Q aP P aQ P Q

− ∆ − ∆ + ∆ += +

∆ + ∆ + ∆ ∆ + + ∆ −

( )( )

2

20 1 04 csc 2

,cot 2

q f fa− ∆ ∆

= +∆ + ∆

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402

where P and Q are two non-zero real constants.

Family 3.2: The periodic form solutions of Eq. (6) (when a – 4bc < 0 and ab 0 or bc 0) are:2

where , and a,b,c are arbitrary constants.

where P and Q are two non-zero real constants and satisfies P – Q > 0.2 2

( )( )

2

21 1 04 csc 2

,tan 2

q f fa− ∆ ∆

= +∆ + ∆

( ) ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( )( )

2

22 1 02 2

2 sec 2 1 sin 2 cos 2 2 sin 2 2.

2 cos 2 2 1 sin 2 2 cos 2

aq f f

a bc a

− ∆ ∆ ± ∆ ∆ + ∆ ∆ ± ∆= +

− ∆ + ∆ ± ∆ ∆ ± ∆

( ) ( ) ( )( )( ) ( ) ( )

2

23 1 02

2 csc 2 sin 2 2 cos 2 2.

2 cos 2 2 sin 2 2

aq f f

bc a a bc

∆ ∆ − ∆ + ∆ ∆ ± ∆= +

− ∆ − ∆ ∆ ±

( )( )

2 2

24 1 02 csc

,2 cot

q f fa− ∆ ∆

= ++ ∆ ∆

( ) ( )( )( ) ( )25 1 0

cosh sinh,

cosh sinha a a

q f fp a a

−= +

+ −

( ) ( )26 1 0,cosh sinh

apq f fp a a

= ++ +

6 ,0 2 2 8

kf aia bc

=+ +

621 2 2 8

kf ia bc

= ±+ +

4 .2 2 8

kx ta bc

= −

+ +

127

1,

f bq

b g−

=+

621 2 2 8

kf ia bc

= ±+ +

4 .2 2 8

kx ta bc

= −

+ +

( ) 1, 2 3 .3,4 2u x t

x t= ±

+( ) 1, 2 3 .3,4 2

u x tx t

= ±+

( ) 1, 2 3 .3,4 2u x t i

x t= ±

−( ) 1, 2 3 .3,4 2

u x t ix t

= ±−

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403

Family 3.3: The soliton and soliton-like solutions of Eq. (6) (when c = 0 and ab 0) are:

where p is an arbitrary constant, and

Family 3.4: The rational function solution (when b 0 and c = a = 0) is:

where g is an arbitrary constant, and 1

Table 1: Comparison between Liu et al. [48] solutions and New solutions

Liu et al. [48] solutions New solutions

i. If C = 1, C = 1, µ = 1, = 2, a = 1 and i. If = 1, g = 0, b = 1, k = –1 and q ( ) = u (x,t)1 2 1 27 3,4

k = -1, from section 3 in example 1 solution becomes: solution q becomes: 27

ii. If C = 1, C = 1, µ = 1, = 2, a = 1 and ii. If = 1, g = 0, b = 1, k = 1 and q ( ) = u (x,t) and1 2 1 27 3,4

k = 1, from section 3 in example 2 solution becomes: solution q becomes: 27

RESULTS AND DISCUSSION In addition, the graphical representations of some

It is significant to state that one of our obtained to Figure 8.solutions is in good harmony with the existing resultswhich are shown in the Table 1. Beyond the table, we Graphical Representations of the Solutions:have constructed new exact traveling wave solutions The graphical illustrations of the solutionsq to q which have not been reported in the previous are depicted in the figures with the aid of1 26

literature. Maple.

obtained traveling wave solutions are shown in Figure 1

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404

Fig. 1: Solitons solutions for Fig. 4: Periodic solutions fora = 4,b = 7, c = 0.05, = 12, k = 0.5 a = 6,b = 7, c = 4, = 9, k = 0.05

Fig. 2: Solitons solutions for Fig. 5: Solitons solutions fora = 3,b = 2, c = 1, = 1, k = 3 a = 7,b = 8, c = 4, = 10, k = 5×k

Fig. 3: Periodic solutions for Fig. 6: Solitons solutions for a = 4,b = 5, c = 5, = 5, k = 0.25 a = 5,b = 4, c = 4, = 2, k = 5×10

1

8

World Appl. Sci. J., 22 (3): 396-407, 2013

405

Fig. 7: Periodic solutions fora = 0,b = 3, c = 0, = 2, k = 0.5,g = 11

Fig. 8: Periodic solutions fora = 0,b = 3, c = 0, = 2, k = 0.5,g = 01

CONCLUSIONS

In this article, we have constructed twenty sevenexact traveling wave solutions including solitons, periodicand rational solutions for the simplified MCH equation viathe (G'/G)-expansion method together with thegeneralized Riccati equation mapping. The traveling wavesolutions are expressed in terms of the hyperbolic, thetrigonometric and the rational functions. The performanceof this method is trustworthy and gives many newsolutions. Moreover, one of our obtained solutions is ingood agreement with the existing results which validatesour other solutions. Therefore, the (G'/G)-expansionmethod together with the generalized Riccati equation can

be further used to solve many nonlinear evolutionequations which frequently arise in various scientific realtime application fields. Further, future directions of theresearch work will be towards solving complicatedapplication problems and resolving difficulties using theaforementioned method.

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