Exact travelling wave solutions of some nonlinear equations by -expansion method

Applied Mathematics and Computation 216 (2010) 2596–2612

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Exact travelling wave solutions of some nonlinear equationsby G0

G

� �-expansion method

Anand Malik, Fakir Chand *, S.C. MishraDepartment of Physics, Kurukshetra University, Kurukshetra 136119, India

a r t i c l e i n f o

Keywords:Nonlinear evolution equationsSolitonsTravelling wave solutions

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.03.103

* Corresponding author.E-mail addresses: [email protected] (A. Malik

a b s t r a c t

Here we find exact solutions of some nonlinear evolution equations within the frame-work of the G0

G

� �-expansion method. Exact solutions of five nonlinear equations of physical

importance viz. the coupled Schrodinger–KdV equation, the coupled nonlinear Reaction–Diffusion equation, the Foam Drainage equation, the Phi-Four equation and the Dodd–Bullough–Mikhailov equation are obtained. These general solutions can be reduced insome standard results derived by some other methods. Two and three dimensional plotsof some of the results are also presented.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

Many problems of solid state physics, fluid mechanics, plasma physics, population dynamics, chemical kinetics, nonlinearoptics etc. are described by nonlinear evolution equations [1,2]. Once a phenomenon is modeled in terms of mathematicalequations, one is generally interested to find exact analytic solutions of such nonlinear equations in order to predict, controland quantify the underlying features of the system under study.

In the past considerable efforts have been made to obtain exact analytical solutions of nonlinear equations and a numberof methods have been developed for obtaining explicit travelling wave solutions of nonlinear evolution equations [1,3–16].

Recently, Wang et al. [17] developed a new technique called G0

G

� �-expansion method for a reliable treatment of nonlinear

wave equations. Thereafter some more applications of this method have also been reported [18–23]. A generalized version ofG0

G

� �-expansion method is also reported recently [24,25].In order to further expand the domain of applications of G0

G

� �-expansion method, in the present study we explore five non-

linear equations of physical interest i.e. the coupled Schrodinger–KdV equation, the coupled nonlinear Reaction–Diffusionequation, the Foam Drainage equation, the Phi-Four equation and the Dodd–Bullough–Mikhailov equation.

The organization of the paper is as follows: in Section 2, a brief description of the G0

G

� �-expansion method for finding the

travelling wave solutions of nonlinear equations is presented. In Section 3 five problems are considered and solved by thismethod. Finally concluding remarks are given in Section 4.

2. The G0G

� �-expansion method

Here we briefly describe the main steps of the G0

G

� �-expansion method. Suppose that a nonlinear partial differential

equation (PDE) is of the form

Pðu;ut ;ux;utt ;uxt ;uxx; . . .Þ ¼ 0; ð1Þ

. All rights reserved.

), [email protected], [email protected] (F. Chand).

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A. Malik et al. / Applied Mathematics and Computation 216 (2010) 2596–2612 2597

where u = u(x, t) is an unknown function and P is polynomial in u = u(x,t) and its partial derivatives, in which higher orderderivatives and nonlinear terms are involved.

Step 1: To find the travelling wave solution of (1), introduce the wave variable n = (x � ct), so that u(x, t) = u(n).Based on this, we use the following changes

@

@t¼ �c

@

@n;

@2

@t2 ¼ c2 @2

@n2 ;@

@x¼ @

@n;

@2

@x2 ¼@2

@n2 ; ð2Þ

and so on for other derivatives. With the help of (2), the PDE (1) changes to an ordinary differential equation (ODE) as

Pðu;un;unn;unnn; . . .Þ ¼ 0; ð3Þ

where un, unn etc. denote derivative of u with respect to n.Now integrate the ODE (3) as many times as possible and set the constants of integration to be zero.Step 2: The solution of (3) can be expressed by a polynomial in G0

G

� �i.e.

uðnÞ ¼ amðG0

GÞm þ am�1ð

G0

GÞm�1 þ � � � ; ð4Þ

where G = G(n) satisfies the second order linear ODE of the form

G00 þ kG0 þ lG ¼ 0; ð5Þ

where am, am�1,. . .,a0, k and l are constants to be determined later and am – 0.The positive integer m can be determined by considering the homogeneous balance between the highest order deriv-atives and nonlinear terms appearing in ODE (3), after using (4).Step 3: Substituting (4) into (3) and using (5), collecting all terms with the same order of G0

G

� �together, and then

equating each coefficient of the resulting polynomial to zero yields a set of algebraic equations for am, am�1,. . .,a0, c, k and l.Step 4: Since the general solutions of (5) have been well known for us, then substituting am, am�1,. . ., a0, c and thegeneral solutions of (5) into (4) we obtain more travelling wave solutions of nonlinear differential equation (1).

� �
After the brief description of the G0
G -expansion method, we now solve some PDEs of physical importance using it.

3. Applications

Here we apply the G0

G

� �-expansion method to find travelling wave solutions for the coupled Schrodinger–KdV equation, the

(2+1)-dimensional coupled nonlinear Reaction–Diffusion equation, the Foam drainage equation, the Phi-Four equation andthe Dodd–Bullough–Mikhailov equation. These equations have many applications in different fields.

3.1. The coupled Schrodinger–KdV equation

Consider the coupled Schrodinger–KdV equation

ıut ¼ uxx þ uv ;v t þ 6vvx þ vxxx ¼ ðjuj2Þx;

ð6Þ

which describes various processes in dusty plasma, such as Langmuir, dust-acoustic wave and electromagnetic waves[13,26,27]. A soliton solution of this equation is obtained by Hase and Satsuma [28].

Consider

u ¼ eıhUðnÞ; v ¼ VðnÞ; h ¼ axþ bt; n ¼ xþ ct; ð7Þ

where a, b and c are constants. On substituting (7) into (6), we find c = 2a, whereas U and V satisfy the following coupledODEs

U00 þ ðb� a2ÞU þ UV ¼ 0; ð8aÞV 000 þ 2aV 0 þ 6VV 0 � ðU2Þ0 ¼ 0: ð8bÞ

On integrating (8b), we obtain

V 00 þ 2aV þ 3V2 � U2 þ c1 ¼ 0; ð9Þ

where c1 is a constant of integration.Considering the homogeneous balance between U

00and UV in (8a) and that between V

00and V2 in (9), we get m1 = m2 = 2.

Thus, from (4), the solutions of (8a) and (9) are written as

2598 A. Malik et al. / Applied Mathematics and Computation 216 (2010) 2596–2612

UðnÞ ¼ a2G0

G

� �2

þ a1G0

G

� �þ a0; a2 – 0; ð10aÞ

VðnÞ ¼ b2G0

G

� �2

þ b1G0

G

� �þ b0; b2 – 0; ð10bÞ

where a0, a1, a2, b0, b1 and b2 are constants to be determined later and G = G(n) satisfies the second order linear ODE (5).Now substituting (10a) and (10b) into (8a) and (9) and equating all terms with the same power of G0

G

� �to zero, we obtain a

set of simultaneous algebraic equations as

6a2 þ a2b2 ¼ 0; ð11aÞ2a1 þ 10ka2 þ a2b1 þ a1b2 ¼ 0; ð11bÞð4k2 þ 8lþ b� a2Þa2 þ 3ka1 þ a1b1 þ a2b0 þ a0b2 ¼ 0; ð11cÞð2lþ k2 þ b� a2Þa1 þ 6lka2 þ a1b0 þ a0b1 ¼ 0; ð11dÞ2l2a2 þ lka1 þ ðb� a2Þa0 þ a0b0 ¼ 0; ð11eÞ

and

6b2 þ 3b22 � a2

2 ¼ 0; ð12aÞ2b1 þ 10kb2 þ 6b2b1 � 2a2a1 ¼ 0; ð12bÞð4k2 þ 8lÞb2 þ 3kb1 þ 6b2b0 þ 3b2

1 � 2a2a0 � a21 þ 2ab2 ¼ 0; ð12cÞ

ð2lþ k2Þb1 þ 6lkb2 þ 6b1b0 � 2a1a0 þ 2ab1 ¼ 0; ð12dÞ2l2b2 þ lkb1 þ 3b2

0 � a20 þ 2ab0 þ c1 ¼ 0: ð12eÞ

On solving the above algebraic equations, we have

a0 ¼ �1ffiffiffi2p ðk2 þ 8lÞ þ 2

5ffiffiffi2p ð3ðb� a2Þ � aÞ

� �; a1 ¼ �6

ffiffiffi2p

k;

a2 ¼ �6ffiffiffi2p

;b0 ¼ �12ðk2 þ 8lÞ þ 1

5ð2ðb� a2Þ þ aÞ

� �; b1 ¼ �6k;

b2 ¼ �6; c1 ¼ �1

25ð3ðb� a2Þ2 � 8a2 � 2aðb� a2ÞÞ;

ðk2 � 4lÞ ¼ �25ð3ðb� a2Þ � aÞ: ð13Þ

Hence in view of the solutions (13), Eqs. (10a) and (10b) become

UðnÞ ¼ �6ffiffiffi2p G0

G

� �2

� 6ffiffiffi2p

kG0

G

� �þ a0; ð14aÞ

VðnÞ ¼ �6G0

G

� �2

� 6kG0

G

� �þ b0; ð14bÞ

whereas a0 and b0 are given in (13). Substituting the general solution of second order linear ODE (5) into (14a) and (14b), weobtain three types of travelling wave solutions of the coupled Schrodinger–KdV equations (8a) and (9) as

(i). When k2 � 4l > 0, we write

UðnÞ ¼ � 3ffiffiffi2p ðk2 � 4lÞ

A1sinhð12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

qÞnþ A2cosh 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

q� �n

A1cosh 12


q� �nþ A2sinhð12


qÞn

0BB@

1CCA

2

� 3ffiffiffi2p k2 þ a0; ð15aÞ

VðnÞ ¼ �32ðk2 � 4lÞ

A1sinhð12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � 4l

qÞnþ A2coshð12


qÞn

A1cosh 12


q� �nþ A2sinh 1

2


q� �n

0BB@

1CCA

2

þ 32

k2 þ b0: ð15bÞ

Finally the travelling wave solutions of (6) are written as


uðnÞ ¼ � 3ffiffiffi2p ðk2 � 4lÞ

A1sinh 12


q� �nþ A2cosh 1

2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA

2

� 3ffiffiffi2p k2 þ a0

0BBB@

1CCCAeıh; ð16aÞ

vðnÞ ¼ �32ðk2 � 4lÞ

A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA

2

þ 32

k2 þ b0; ð16bÞ

where A1 and A2 are arbitrary constants.In particular, if A1 – 0, A2 = 0, k > 0, l = 0, then u(n) and v(n) become

uðnÞ ¼ � 3ffiffiffi2p k2sech2 1

2knþ a0

� �eıh; ð17aÞ

vðnÞ ¼ 32

k2sech2 12

kn

� �þ b0; ð17bÞ

which are periodic and solitonic wave solution of the coupled Schrodinger–KdV equation and which are similar to the resultsobtained using tanh-method in [13,28]. The diagrams of these solutions are given in Figs. 1.1–1.6 for some particular valuesof constants.

(ii). When k2 � 4l < 0, one obtains the solutions as

uðnÞ ¼ � 3ffiffiffi2p ð4l� k2Þ

�A1sin 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4l� k2

q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA

2


0BBB@

1CCCAeıh; ð18aÞ

vðnÞ ¼ �32ð4l� k2Þ

�A1sin 12


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA

2

þ 32

k2 þ b0: ð18bÞ

(iii). When k2 � 4l = 0, the results are given as

Fig. 1.1. 3D Plots of real part of Eq. (17a) for a = 1, b = 2.

Fig. 1.2. 3D Plots of imaginary part of Eq. (17a) for a = 1, b = 2.

Fig. 1.3. 2D plots: periodic solutions of real part of Eq. (17a) for a = 1, b = 2 and t = 0.02.

Fig. 1.4. 2D plots: periodic solutions of imaginary part of Eq. (17a) for a = 1, b = 2 and t = 0.02.


Fig. 1.5. 3D plot for a = 1, b = 2 and t = 0.02 of Eq. (17b).

Fig. 1.6. 2D plot for a = 1, b = 2 and t = 0.02 of Eq. (17b).


uðnÞ ¼ �6ffiffiffi2p A2

A1 þ A2n

� �2


!eıh; ð19Þ

vðnÞ ¼ �6A2

A1 þ A2n

� �2

þ 32

k2 þ b0: ð20Þ

These are the travelling wave solutions of coupled Schrodinger–KdV equation under different assumption.

3.2. The (2+1)-dimensional coupled nonlinear Reaction–Diffusion equation

The (2+1)-dimensional coupled nonlinear extension of Reaction–Diffusion equation (CNLERD) is given by [29,30]

ut þ uxy �wu ¼ 0;v t � vxy þwv ¼ 0;wx þ ðuvÞy ¼ 0:

ð21Þ

where u, v and w are physical observables and subscripts denote partial differentiations. A physical application of (21) hasbeen pointed out by Duan et al. [31] while presenting (21) as a corresponding geometric equivalent (2+1)-dimensional


CNLERD equation of the integrable (2+1)-dimensional (modified) Heisenberg ferromagnet model. The complete integrabilityof this equation, using the technique of Painlevè-analysis, is investigated in [32].

Using the wave variable n = x + y � ct, the system of PDEs (21) is carried to a system of ODEs

� cu0 þ u00 �wu ¼ 0;� cv 0 � v 00 þwv ¼ 0;

w0 þ ðuvÞ0 ¼ 0:

ð22Þ

Integrating the third equation in the system and neglecting constant of integration, we find

w ¼ �uv : ð23Þ

Substituting (23) into the first and second equations of the system, we find

u00 � cu0 þ u2v ¼ 0;

v 00 þ cv 0 þ uv2 ¼ 0:ð24Þ

On balancing u00

with u2v and v00 with v2u in (24), one obtains

m1 ¼ m2 ¼ 1: ð25Þ

Here, we suppose that

uðnÞ ¼ a1G0

G

� �þ a0; a1 – 0; ð26aÞ

vðnÞ ¼ b1G0

G

� �þ b0; b1 – 0; ð26bÞ

where G = G(n) satisfies the second order linear ODE (5) where a1, a0, b1 and b0 are constants to be determined later.By substituting (26a) and (26b) into (24) and collecting all terms with the same power of G0

G

� �together, the left-hand sides

of (24) are converted into the polynomials in G0

G

� �. Equating each coefficient of the polynomials to zero, yield a set of simul-

taneous algebraic equations as

2a1 þ a21b1 ¼ 0; ð27aÞ

ð3kþ cÞa1 þ 2a0a1b1 þ a21b0 ¼ 0; ð27bÞ

ð2lþ k2 þ kcÞa1 þ a20b1 þ 2a0a1b0 ¼ 0; ð27cÞ

ðkþ cÞla1 þ a20b0 ¼ 0; ð27dÞ

and

2b1 þ a1b21 ¼ 0; ð28aÞ

ð3k� cÞb1 þ 2a1b0b1 þ a0b21 ¼ 0; ð28bÞ

ð2lþ k2 � kcÞb1 þ a1b20 þ 2a0b0b1 ¼ 0; ð28cÞ

ðk� cÞlb1 þ a0b20 ¼ 0: ð28dÞ

On solving the above set of algebraic equations, we have

a1 ¼ �2b1; a0 ¼ �

1b1ðkþ cÞ; b1 ¼ b1;

b0 ¼12ðk� cÞb1; c ¼ �


q;

ð29Þ

where k, l and b1 are arbitrary constants. Now, substituting (29) into (26a) and (26b), we get the solutions of (24) as

uðnÞ ¼ � 2b1

G0

G

� �� 1

b1ðkþ cÞ; ð30aÞ

vðnÞ ¼ b1G0

G

� �þ 1

2ðk� cÞb1; ð30bÞ

wðnÞ ¼ 2G0

G

� �2

þ 2kG0

G

� �þ 1

2ðk2 � c2Þ: ð30cÞ

Inserting the general solution of second order linear ODE (5) into (30a)–(30c), we derive three types of travelling wave solu-tions of the CNLERD equation, which are given as

(i). When k2 � 4l > 0, the solutions become


uðnÞ ¼ �


qb1

A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA �


qb1

; ð31aÞ

vðnÞ ¼


q2

b1

A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA �


q2

b1; ð31bÞ

wðnÞ ¼ ðk2 � 4lÞ

2

A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA

2

� ðk2 � 4lÞ

2: ð31cÞ

Note that, if A1 – 0, A2 = 0, k > 0, l = 0, then u, v and w become

uðnÞ ¼ kb1�1� tanh

k2

n

� �; ð32aÞ

vðnÞ ¼ k2

b1 �1þ tanhk2

n

� �; ð32bÞ

wðnÞ ¼ � k2

21� tanh2 k

2n

� �; ð32cÞ

which are the solitary wave solution of CNLERD equation [30]. The plots of these results are shown in Figs. 2.1–2.6.(ii). When k2 � 4l < 0, we write

uðnÞ ¼ �


qb1

�A1sin 12


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA �


qb1

; ð33aÞ

vðnÞ ¼


q2

b1

�A1sin 12


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA �


q2

b1; ð33bÞ

wðnÞ ¼ ð4l� k2Þ2

�A1sin 1

2


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA

2

� ðk2 � 4lÞ

2: ð33cÞ

Fig. 2.1. 3D plot of Eq. (32a) when b 1 = 1, k = 1 and t = 0.02.

Fig. 2.2. 3D plot of Eq. (32b) when b 1 = 1, k = 1 and t = 0.02.

Fig. 2.3. 3D plot of Eq. (32c) when b 1 = 1, k = 1 and t = 0.02.


(iii). When k2 � 4l = 0, we derive the solution as

uðnÞ ¼ � 2b1

A2

A1 þ A2n

� �; ð34aÞ

vðnÞ ¼ b1A2

A1 þ A2n

� �; ð34bÞ

wðnÞ ¼ 2A2

A1 þ A2n

� �2

; ð34cÞ

where A1 and A2 are arbitrary constants.

Fig. 2.4. 2D plot of Eq. (32a) for b 1 = 1, k = 1, t = 0.02 and y = 0.

Fig. 2.5. 2D plot of Eq. (32b) for b 1 = 1, k = 1, t = 0.02 and y = 0.


3.3. The Foam Drainage equation

The Foam Drainage equation [30,34] is given as

ut þ u2 � 12

ffiffiffiup

ux

� �x

¼ 0; ð35Þ

where x and t are scaled position and time coordinates respectively. Foam is central to a number of everyday activities, bothnatural and industrial. As such foam has been of great interest for academic research. In the process industries, foam can be adesirable and even essential element of a process. An example is in the case of froth flotation separation of minerals and coal[33]. Foaming occurs in many distillation and absorption processes. Foams are very important in many technological pro-cesses and applications. Their properties are subjected to intensive investigational efforts from both practical developersand scientific researchers [34].

Using the wave variable n = k (x + ct), (35) is carried to an ODE

cku0 þ kðu2 � k2

ffiffiffiup

u0Þ0 ¼ 0: ð36Þ

On integrating (36) with respect to n and considering the integration constant zero, we obtain

Fig. 2.6. 2D plot of Eq. (32c) for b 1 = 1,k = 1, t = 0.02 and y = 0.


ckuþ k u2 � k2

ffiffiffiup

u0� �

¼ 0: ð37Þ

Here, after using the transformation u(n) = v2(n), (37) becomes

c þ v2 � kv 0 ¼ 0: ð38Þ
Now on balancing v0 with v2 in (38), we get m = 1.
Here, we suppose that

vðnÞ ¼ a1ðG0

GÞ þ a0; a1 – 0; ð39Þ

where a1 and a0 are constants to be determined later and G = G(n) satisfies the second order linear ODE (5).By substituting (39) into (38) and collecting all terms with the same power of G0

G

� �together, the left-hand sides of (38) are

converted into the polynomials in G0

G

� �. Equating each coefficient of the polynomials to zero, yields a set of simultaneous alge-

braic equations as

� ka1 � a21 ¼ 0; ð40aÞ

� kka1 � 2a1a0 ¼ 0; ð40bÞ� kla1 � a2

0 � c ¼ 0: ð40cÞ

The solutions of the above set of algebraic equations are obtained as

a1 ¼ �k; a0 ¼ �k2

k; c ¼ � k2

4ðk2 � 4lÞ; ð41Þ

where k, l and k are arbitrary constants. Now, inserting (41) into (39), we get the solution of (38) as

vðnÞ ¼ �kG0

G

� �� k

2k: ð42Þ

Substituting the general solution of second order linear ODE (5) into (42), we have three types of travelling wave solutions ofthe Foam Drainage equation as

(i). For k2 � 4l > 0

vðnÞ ¼ � k2


q A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA; ð43aÞ

uðnÞ ¼ k2

4ðk2 � 4lÞ

A1sinh 12



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA

2

: ð43bÞ


In particular, if A1 – 0, A2 = 0, k > 0, l = 0, then v and u become

vðnÞ ¼ � kk2

tanhk2

n; ð44Þ

uðnÞ ¼ k2k2

4tanh2 k

2n; ð45Þ

which are the solitary wave solution of (38) and the Foam Drainage equation [30,34]. The plots of these solutions are shownin Figs. 3.1 and 3.2.

(ii). For k2 � 4l < 0

vðnÞ ¼ � k2


q �A1sin 12


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA; ð46aÞ

uðnÞ ¼ k2

4ð4l� k2Þ

�A1sin 12


q� �nþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA

2

: ð46bÞ

(iii). For k2 � 4l = 0

vðnÞ ¼ �kð A2

A1 þ A2nÞ; ð47aÞ

uðnÞ ¼ k2ð A2

A1 þ A2nÞ2; ð47bÞ


3.4. The Phi-Four equation

Consider the Phi-Four equation [12,35]

utt � auxx � u� u3 ¼ 0: ð48Þ

This equation arises in the Quantum Field Theory in the study of quartic interaction theory. Recently, the Phi-Four equationwas also studied by Sassaman and Biswas [36] using soliton perturbation theory.

Using the wave variable n = x � ct, Eq. (48) becomes an ODE

Fig. 3.1. 3D plot of Eq. (45) when k = 1, k = 1.

Fig. 3.2. 2D plot of Eq. (45) for k = 1, k = 1 and t = 0.02.


ðc2 � aÞu00 þ u3 � u ¼ 0: ð49Þ

Now, balancing u00

with u3 in (49), we get m = 1. Now suppose that

uðnÞ ¼ a1ðG0

GÞ þ a0; a1 – 0; ð50Þ

where G = G(n) satisfies the second order linear ODE (5) and a1 and a0 are constants to be determined later.Again (50) and (49) yield a set of simultaneous algebraic equations as

2ðc2 � aÞa1 þ a31 ¼ 0; ð51aÞ

3ðc2 � aÞka1 þ 3a21a0 ¼ 0; ð51bÞ

ð2lþ k2Þðc2 � aÞa1 þ 3a1a20 � a1 ¼ 0; ð51cÞ

ðc2 � aÞlka1 þ a30 � a0 ¼ 0: ð51dÞ

The solutions of the above equations is written as

a1 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ða� c2Þ

q; a0 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ða� c2Þ

q k2; ðk2 � 4lÞ ¼ 2

a� c2 : ð52Þ

Now, substituting (52) into (50) and then with the help of second order linear ODE (5), we obtain three different travellingwave solutions of the Phi-Four equation (48) as

(i). When k2 � 4l > 0

uðnÞ ¼ �A1sinh 1

2



2


q� �n

A1cosh 12



2


q� �n

0BB@

1CCA; ð53Þ

In particular, if A1 – 0, A2 = 0, k > 0, l = 0, then u becomes

uðnÞ ¼ �tanhk2

n: ð54Þ

However, if A1 = 0, A2 –, k > 0, l = 0, then u becomes

uðnÞ ¼ �cothk2

n; ð55Þ

which are the solitary wave solution of the Phi-Four equation (48) and shown in Figs. 4.1 and 4.2.(ii). When k2 � 4l < 0

Fig. 4.1. 3D plot of Eq. (54) for a = 3, k = 1.

Fig. 4.2. 2D plot of Eq. (54) for a = 3, k = 1 and t = 0.02.


uðnÞ ¼ ��A1sinð12


qÞnþ A2cos 1

2


q� �n

A1cos 12


q� �nþ A2sin 1

2


q� �n

0BB@

1CCA; ð56Þ


3.5. The Dodd–Bullough–Mikhailov equation

The Dodd–Bullough–Mikhailov (DBM) equation reads [13,37]

uxt þ aeu þ be�2u ¼ 0; ð57Þ

which becomes Liouville equation when b = 0.We make transformations u = ln v, v = V (n), n = x + ct. Then (57) becomes

cVV 00 � cV 02 þ aV3 þ b ¼ 0: ð58Þ


Considering the homogeneous balance between VV00

and V3, we get m = 2.We simply suppose that the solution of (58) is of the form

VðnÞ ¼ a2ðG0

GÞ2 þ a0; a2 – 0; ð59Þ

where G = G(n) satisfies the second order linear ODE

G00 þ lG ¼ 0; ð60Þ

where a1, a0 and l are constants to be determined later.Again (59) and (58) yield a set of simultaneous algebraic equations

2ca22 þ aa3

2 ¼ 0; ð61aÞ6ca2a0 þ 3aa2

2a0 ¼ 0; ð61bÞð8la2a0 � 2l2a2

2Þc þ 3aa2a20 ¼ 0; ð61cÞ

2cl2a2a0 þ aa30 þ b ¼ 0; ð61dÞ

whose solutions are given by

a2 ¼ �2ca; a0 ¼ �

2lc3a

; b ¼ �64l3c3

3a2 ; ð62aÞ

a2 ¼ �2ca; a0 ¼ �

2lca

; b ¼ 0: ð62bÞ

The results in (62a) give solution of DBM equation (b – 0) and in (62b) give solution of Liouville equation (b = 0).The solutions of (58) for two cases ((62a) and (62b)) are written as

V1ðnÞ ¼ �2ca

G0

G

� �2

� 2lc3a

; ð63aÞ

V2ðnÞ ¼ �2ca

G0

G

� �2

� 2lca

: ð63bÞ

CASE (a): Solution of DBM when b – 0For this particular case, substituting the general solution of second order linear ODE (60) into (63a), we obtain three dif-

ferent travelling wave solutions of DBM equation under different conditions on l.

(i). When l < 0,

V1ðnÞ ¼2lca

A1sinhffiffiffiffiffiffiffi�lp

nþ A2coshffiffiffiffiffiffiffi�lp

nA1cosh

ffiffiffiffiffiffiffi�lpnþ A2sinh

ffiffiffiffiffiffiffi�lpn

� �2

� 2lc3a

; ð64aÞ

u1ðnÞ ¼ ln2lca



nA1cosh



� �2

� 2lc3a

!: ð64bÞ

In particular, if A1 – 0, A2 = 0, then u1 becomes

uðnÞ ¼ ln2lc3að3tanh2 ffiffiffiffiffiffiffi�l

pn� 1Þ

� �: ð65Þ

However, if A1 = 0, A2 –, then u1 will be

uðnÞ ¼ ln2lc3að3coth2 ffiffiffiffiffiffiffi�l

pn� 1Þ

� �: ð66Þ

Note that the solutions given in (65) and (66) are similar to the solitary wave solution of the DBM equation obtained in [13]using extended tanh-method.

(ii). When l > 0

V1ðnÞ ¼ �2lca

�A1sinffiffiffiffilp nþ A2cos

ffiffiffiffilp nA1cos

ffiffiffiffilp nþ A2sinffiffiffiffilp n

� �2

� 2lc3a

; ð67Þ

u1ðnÞ ¼ ln �2lca




� �2

� 2lc3a

!: ð68Þ

(iii). When l = 0


V1ðnÞ ¼ �2ca

A2

A1 þ A2n

� �2

; ð69aÞ

u1ðnÞ ¼ ln �2ca

A2

A1 þ A2n

� �2 !

; ð69bÞ


CASE (b): When b = 0Similarly, we obtain the travelling wave solution of DBM (or Liouville) equation for different conditions on l as

(i). When l < 0

u2ðnÞ ¼ ln2lca



nA1cosh



� �2

� 2lca

!: ð70Þ

In particular, if A1 – 0, A2 = 0, then u1 becomes

uðnÞ ¼ ln �2lca

sech2 ffiffiffiffiffiffiffi�lp

n

� �: ð71Þ

However, if A1 = 0, A2 –, then u1 becomes

uðnÞ ¼ ln2lca

cosech2 ffiffiffiffiffiffiffi�lp

n

� �; ð72Þ

which are the solitary wave solution of the Liouville equation [13].(ii). When l > 0

u2ðnÞ ¼ ln �2lca




� �2

� 2lca

!: ð73Þ

(iii). When l = 0

u2ðnÞ ¼ ln �2ca

A2

A1 þ A2n

� �2 !

; ð74Þ


4. Conclusion

With a view to further expand the domain of applications of the G0

G

� �-expansion method, in this work we obtained exact

solutions of some nonlinear evolution equations namely the coupled Schrodinger–KdV equation, the (2+1)-dimensional cou-pled nonlinear Reaction–Diffusion equation, the Foam Drainage equation, the Phi-Four equation and the Dodd–Bullough–Mikhailov equation. The general travelling wave solutions can give solitonic or periodic solutions under different parametricrestrictions. The two and three dimensional plots of some the results prove the veracity of the analytic solutions. It is inter-esting to note that from the general results, one can easily recover solutions which are obtained from others methods. Thisdirect and concise method can further be used to explore more applications.

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Exact travelling wave solutions of some nonlinear equations by -expansion method