Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations

Applied Mathematics and Computation 169 (2005) 639–656

www.elsevier.com/locate/amc

Travelling wave solutions of generalizedforms of Burgers, Burgers–KdV and

Burgers–Huxley equations

Abdul-Majid Wazwaz

Department of Mathematics and Computer Science, Saint Xavier University,

3700 West 103rd Street, Chicago, IL 60655, USA

Abstract

In this work, exact travelling wave solutions of generalized forms of Burgers, Bur-

gers–KdV and Burgers–Huxley equations are obtained. The analysis rests mainly on

the standard tanh method. The work emphasizes the need for a transformation formula

for the case where the parameter M is non-integer. The approach can be used in a vari-

ety of many types of nonlinearity.

� 2004 Elsevier Inc. All rights reserved.

Keywords: The tanh method; Burgers equation; Burgers–KdV equation; Burgers–Huxley equation;

Huxley equation; Travelling wave solutions

1. Introduction

It is known that many phenomena in scientific fields can be described by

nonlinear partial differential equations [1–5]. Typical equations are the KdV

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.09.081

E-mail address: [email protected]

mailto:[email protected]

640 A.-M. Wazwaz / Appl. Math. Comput. 169 (2005) 639–656

equation, Burgers equation, Boussinesq equation and many others. Many

problems, such as the KdV equation, involve dispersion, other problems, such

as Burgers equation, involve dissipation, whereas other problems involve both

dispersion and dissipation such as the Burgers–KdV equation [2]. Typical

problems of dispersion and dissipation are the flow of liquids containing gas

bubbles and the propagation of waves on an elastic tube field with a viscousfluid [2].

A variety of powerful methods, such as Backlund transformation, the in-

verse scattering method, bilinear transformation, the tanh–sech method [6–

9], the sine–cosine method [10–20], extended tanh method, pseudo spectral

method, the homogeneous balance method, and the trial function [3] were used

to investigate nonlinear dispersive and dissipative problems.

Our first interest in the present work being in implementing the tanh method

to stress its power in handling nonlinear equations so that one can apply it tomodels of various types of nonlinearity. The next interest is in the determina-

tion of exact travelling wave solutions for generalized forms of Burgers equa-

tion, Burgers–KdV equation, and Burgers–Huxley equation. Searching for

exact solutions of nonlinear problems has attracted a considerable amount

of research work where computer symbolic systems facilitate the computa-

tional work.

The parameter M, of the power series in tanh of the tanh method, plays a

major role in this method in that it should be a positive integer to derive aclosed form analytic solution. However, for non-integer values of M, we usu-

ally use a transformation formula to overcome this difficulty and to obtain

exact travelling wave solutions.

As stated before, we aim to obtain travelling wave solutions for generalized

forms B(n, 1) and B(n,n) of the Burgers� equation

ut þ aðunÞx þ buxx ¼ 0; n > 1; a; b 6¼ 0; ð1Þ

ut þ aðunÞx þ bðunÞxx ¼ 0; n > 1; a; b 6¼ 0; ð2Þ

and for generalized forms of the Burgers–KdV equation BKdV(n, 1) and

BKdV(n,n) given by

ut þ aðunÞx � buxxx þ uxx ¼ 0; n > 1; a; b 6¼ 0; ð3Þ

ut þ aðunÞx � buxxx þ ðunÞxx ¼ 0; n > 1; a; b 6¼ 0; ð4Þ

respectively. In addition we aim to study the Huxley equation given by

ut � auxx � uðk � unÞðun � 1Þ ¼ 0; ð5Þand the Burgers–Huxley equation

ut � auxx þ bux � uðk � unÞðun � 1Þ ¼ 0; ð6Þ

A.-M. Wazwaz / Appl. Math. Comput. 169 (2005) 639–656 641

respectively, to formally derive more exact travelling wave solutions. Eqs. (1)–

(6) provide us with the means to meet the primary goals of this work.

In what follows, the tanh method will be reviewed briefly because details can

be found in [6–9].

2. Analysis of the method

The tanh method is a powerful solution method for the computation of

exact traveling wave solutions [6–9]. Various extension forms of the tanh

method have been developed. A power series in tanh was used as an ansatz

to obtain analytical solutions of traveling wave type of certain nonlinear evo-

lution equations.

The wave variable n = (x � ct) or n = x + y � ct carries a nonlinear PDE

P ðu; ut; ux; uxx; uxxx; . . .Þ ¼ 0; ð7Þ

to a nonlinear ODE

Qðu; u0; u00; u000; . . .Þ ¼ 0: ð8ÞEq. (8) is then integrated as long as all terms contain derivatives where integra-

tion constants are neglected.

We then introduce a new independent variable

Y ¼ tanhðlnÞ; ð9Þthat leads to the change of derivatives:

d

dn¼ lð1� Y 2Þ d

dY;

d2

dn2¼ l2ð1� Y 2Þ �2Y

d

dYþ ð1� Y 2Þ d2

dY 2

� �;

ð10Þ

where other derivatives can be derived in a similar manner.

We then propose the following series expansion

uðlnÞ ¼ SðY Þ ¼XMk¼0

akY k; ð11Þ

where M is a positive integer, in most cases, that will be determined. Substitut-ing (10) and (11) into the simplified ODE yields an equation in powers of Y.

To determine the parameter M, we usually balance the linear terms of high-

est order in the resulting equation with the highest order nonlinear terms. With

M determined, we collect all coefficients of powers of Y in the resulting equa-

tion where these coefficients have to vanish. This will give a system of algebraic

equations involving the parameters ak, (k = 0, . . . ,M), l, and c. Having


determined these parameters, knowing that M is a positive integer in most

cases, and using (11) we obtain an analytic solution u(x, t) in a closed form.

For non-integer values ofM, appropriate transformation formulae will be used

so that an integer value can be obtained. This will be introduced in the forth-

coming well-known problems that will be studied.

3. The Burgers equation

In this section the generalized Burgers equations B(n, 1) and B(n,n) will be

investigated by using the tanh method.

3.1. The B(n,1) Burgers equation

We first consider the generalized Burgers equation B(n, 1)

ut þ aðunÞx þ buxx ¼ 0; n > 1: ð12Þ

The wave variable n = x � ct carries Eq. (12) to

�cu0 þ aðunÞ0 þ bu00 ¼ 0; ð13Þ

where by integration we get

�cuþ aun þ bu0 ¼ 0: ð14Þ

Balancing u 0 with un gives

2þM � 1 ¼ nM ; ð15Þso that

M ¼ 1

n� 1; n > 1: ð16Þ

A necessary condition for obtaining a closed form analytic solution requires

that M be a positive integer. It is normal to use the transformation

u ¼ v1

n�1; ð17Þthat carries (14) to

�cðn� 1Þvþ aðn� 1Þv2 þ bv0 ¼ 0: ð18ÞBalancing v 0 with v2 we find

2þM � 1 ¼ 2M ; ð19Þso that

M ¼ 1: ð20Þ


Using the tanh method we set

uðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y : ð21Þ

Substituting (21) into (18), and collecting the coefficients of Y gives the system

of algebraic equations for a0, a1 and l:

Y 2 coeff:: �aa21 þ ana21 � bla1 ¼ 0;

Y 1 coeff:: �2aa0a1 � cna1 þ ca1 þ 2ana0a1 ¼ 0;

Y 0 coeff:: �cna0 þ ca0 þ ana20 þ bla1 � aa20 ¼ 0:

ð22Þ

Solving this system gives

a0 ¼c2a

;

a1 ¼c2a

;

l ¼ cðn� 1Þ2b

; a; b 6¼ 0:

ð23Þ

where c is left as a free parameter. Recalling that u ¼ v1

n�1, and using (23), the

kink solitons solutions

uðx; tÞ ¼ c2a

1þ tanhcðn� 1Þ

2bðx� ctÞ

� �� 1n�1

; ð24Þ

and

uðx; tÞ ¼ c2a

1þ cothcðn� 1Þ

2bðx� ctÞ

� �� 1n�1

; ð25Þ

follow immediately.

3.2. The B(�n,1) Burgers equation

We next consider the generalized Burgers equation B(�n, 1)

ut þ aðu�nÞx þ buxx ¼ 0; n > 1: ð26Þ

Proceeding as before we find

M ¼ � 1

nþ 1; ð27Þ

so that we use the transformation

uðx; tÞ ¼ v�1

nþ1: ð28Þ


Following the discussion presented before we get

a0 ¼c2a

;

a1 ¼ � c2a

;

l ¼ cðnþ 1Þ2b

; a; b 6¼ 0:

ð29Þ

where c is left as a free parameter. Recalling that u ¼ v�1

nþ1, the solutions

uðx; tÞ ¼ 1

c2a 1� tanh cðnþ1Þ

2b ðx� ctÞh i� �n o 1

nþ1

; ð30Þ

and

uðx; tÞ ¼ 1

c2a 1� coth cðnþ1Þ

2b ðx� ctÞh i� �n o 1

nþ1

; ð31Þ

are readily obtained for the B(�n, 1) equation given in (12).

3.3. The B(n,n) equation

We consider the generalized Burgers equation B(n,n)

ut þ aðunÞx þ bðunÞxx ¼ 0; n > 1: ð32Þ

The wave variable n = x � ct carries Eq. (32) to

�cu0 þ aðunÞ0 þ bðunÞ00 ¼ 0; ð33Þ

where by integration we get

�cuþ aun þ bðunÞ0 ¼ 0; ð34Þor equivalently

�cuþ aun þ bnun�1u0 ¼ 0; ð35ÞBalancing (un�1u 0) with u gives

ðn� 1ÞM þ 2þM � 1 ¼ M ; ð36Þso that

M ¼ 1

1� n; n > 1: ð37Þ

It is normal to use the transformation

u ¼ v1

1�n; ð38Þ


that carries (35) to

�cð1� nÞv2 þ að1� nÞvþ bnv0 ¼ 0: ð39ÞBalancing v 0 with v2 we find

2þM � 1 ¼ 2M ; ð40Þso that

M ¼ 1: ð41ÞUsing the tanh method we set

uðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y : ð42ÞSubstituting (42) into (39), and collecting the coefficients of Y gives the system

of algebraic equations for a0, a1 and l:

Y 2 coeff:: cna21 � ca21 � bnla1 ¼ 0;

Y 1 coeff:: 2cna0a1 � 2ca0a1 þ aa1 � ana1 ¼ 0;

Y 0 coeff:: �ca20 þ aa0 � ana0 þ bnla1 þ cna20 ¼ 0:

ð43Þ

Solving this system gives

a0 ¼a2c

;

a1 ¼a2c

;

l ¼ aðn� 1Þ2bn

; a; b 6¼ 0;

ð44Þ

where c is left as a free parameter. Recalling that u ¼ v1

1�n, or u ¼ v�1

n�1 and

using (44), the travelling wave solutions

uðx; tÞ ¼ 1

a2c 1þ tanh aðn�1Þ

2bn ðx� ctÞh i� �n o 1

n�1

; ð45Þ

and

uðx; tÞ ¼ 1

a2c 1þ coth aðn�1Þ

2bn ðx� ctÞh i� �n o 1

n�1

; ð46Þ

are readily obtained.

3.4. The B(�n,�n) Burgers equation

The generalized Burgers equation B(�n,�n) is given by

ut þ aðu�nÞx þ bðu�nÞxx ¼ 0; n > 1: ð47Þ


Proceeding as before we find

M ¼ 1

nþ 1; ð48Þ

so that we use the transformation

uðx; tÞ ¼ v1

nþ1: ð49ÞIn a manner parallel to the preceding analysis we find

a0 ¼a2c

;

a1 ¼a2c

;

l ¼ aðnþ 1Þ2b

; a; b 6¼ 0:

ð50Þ

where c5 0 is left as a free parameter. Recalling that u ¼ v1

nþ1, the kink solitons

solutions

uðx; tÞ ¼ a2c

1þ tanhaðnþ 1Þ2bn

ðx� ctÞ� �� 1

nþ1

; ð51Þ

and

uðx; tÞ ¼ a2c

1þ cothaðnþ 1Þ2bn

ðx� ctÞ� �� 1

nþ1

; ð52Þ

follow immediately for the B(�n,�n) equation given in (47).

4. The Burgers–KdV equation

Following the analysis presented before, the generalized Burgers–KdVequation BKdV(n, 1) and BKdV(n,n) will be investigated by using the tanh

method.

4.1. The BKdV(n,1) equation

We will derive exact travelling wave solutions to the BKdV(n, 1)

ut þ aðunÞx � buxxx þ uxx ¼ 0; n > 1; a; b 6¼ 0; n > 1: ð53Þ

Using the wave variable n = x � ct, and integrating the resulting equation, Eq.

(53) will be carried to

�cuþ aun � bu00 þ u0 ¼ 0: ð54Þ


Balancing u00 with un gives

4þM � 2 ¼ nM ; ð55Þ

so that

M ¼ 2

n� 1: ð56Þ

As stated before M must be a positive integer to obtain a closed form solution.

To get such a solution we should use the transformation formula

uðx; tÞ ¼ v2

n�1ðx; tÞ: ð57Þ

into (54) to get

� cðn� 1Þ2v2 þ aðn� 1Þ2v40� 2bðn� 1Þvv00 � 2bð3� nÞðv0Þ2

þ 2ðn� 1Þvv0 ¼ 0: ð58Þ

Balancing vv00 with v4 gives

M þ 4þM � 2 ¼ 4M ; ð59Þ

so that

M ¼ 1: ð60Þ

This means that we can set

vðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y ; ð61Þ

to express the solution v(x, t). Substituting (61) into the reduced ODE (58) and

collecting the coefficients of Y gives the system of algebraic equations for a0, a1,c, and l:

an2a41 � 2ana41 þ aa41 � 2bl2a21n� 2bl2a21 ¼ 0;

2la21 � 2la21n� 4bl2a1na0 � 8ana0a31 þ 4an2a0a31

þ 4aa0a31 þ 4bl2a1a0 ¼ 0;

� ca21 þ 6aa20a21 � 12ana20a

21 þ 2cna21 � cn2a21 þ 6an2a20a

21 þ 8bl2a21

þ 2la1a0 � 2la1na0 ¼ 0;

4an2a30a1 � 2ca0a1 � 2cn2a0a1 þ 4bl2a1na0 � 8ana30a1 � 4bl2a1a0

þ 4cna0a1 þ 2la21nþ 4aa30a1 � 2la21 ¼ 0;

� cn2a20 � ca20 þ an2a40 þ 2cna20 þ 2la1na0 � 2la1a0 þ 2bl2a21n

� 6bl2a21 þ aa40 � 2ana40 ¼ 0:

ð62Þ


Solving the last system gives two sets of solutions

a0 ¼ � 1

nþ 3

ffiffiffiffiffiffiffiffiffiffiffinþ 1

2ab

r;

a1 ¼1

nþ 3


2ab

r;

c ¼ 2ðnþ 1Þbðnþ 3Þ2

;

l ¼ � n� 1

2bðnþ 3Þ :

ð63Þ

Noting that u ¼ v2

n�1 and using (63) we find two sets of exact travelling wave

solutions given by

uðx; tÞ ¼ � 1

nþ 3


2ab

r1þ tanh

n� 1

2bðnþ 3Þ x� 2ðnþ 1Þbðnþ 3Þ2

t

!" # !( ) 2n�1

;

ð64Þ

uðx; tÞ ¼ � 1

nþ 3


2ab

r1þ coth

n� 1

2bðnþ 3Þ x� 2ðnþ 1Þbðnþ 3Þ2

t

!" # !( ) 2n�1

:

ð65Þ

4.2. The BKdV(n,n) equation

We will derive exact travelling wave solutions to the BKdV(n, 1)

ut þ aðunÞx � buxxx þ ðunÞxx ¼ 0; n > 1; a; b 6¼ 0; n > 1: ð66ÞUsing the wave variable n = x � ct into Eq. (66) and integrating the resulting

equation we obtain

�cuþ aun � bu00 þ ðunÞ0 ¼ 0; ð67Þor equivalently

�cuþ aun � bu00 þ nun�1u0 ¼ 0; ð68ÞBalancing u00 with un�1u 0 gives

4þM � 2 ¼ ðn� 1ÞM þ 2þM � 1; ð69Þso that

M ¼ 1

n� 1: ð70Þ


Because M must be a positive integer to obtain a closed form solution, we use

the transformation formula

uðx; tÞ ¼ v1

n�1ðx; tÞ: ð71Þinto (68) to get

�cðn� 1Þ2v2 þ aðn� 1Þ2v3 � bðn� 1Þvv00 � bð2� nÞðv0Þ2 þ nðn� 1Þvv0 ¼ 0;

ð72ÞBalancing vv00 with v2v 0 gives

M þ 4þM � 2 ¼ M þ 2þM � 1; ð73Þso that

M ¼ 1: ð74ÞHowever, balancing vv00 with v3 gives

M þ 4þM � 2 ¼ 3M ; ð75Þso that

M ¼ 2: ð76ÞThis means that, For M = 1, we can set

vðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y ; ð77Þto express the solution v(x, t). Substituting (77) into the reduced ODE (72) and

collecting the coefficients of Y gives the system of algebraic equations for a0, a1,

c, and l:

nla31 � bl2a21n� n2la31 ¼ 0;

2nla21a0 � 2ana31 � 2bl2a1na0 þ 2bl2a1a0 � 2n2la21a0

þ aa31 þ an2a31 ¼ 0;

2bl2a21 þ 3an2a0a21 � nla31 � 6ana0a21 � ca21 þ 3aa0a21 � n2la1a20 � cn2a21

þ n2la31 þ 2cna21 þ nla1a20 ¼ 0;

3aa20a1 � 2bl2a1a0 þ 4cna0a1 þ 2bl2a1na0 þ 2n2la21a0 þ 3an2a20a1

� 2ca0a1 � 6ana20a1 � 2cn2a0a1 � 2nla21a0 ¼ 0;

� cn2a20 � ca20 þ aa30 þ bl2a21nþ n2la1a20 � 2bl2a21 � nla1a20

� 2ana30 þ an2a30 þ 2cna20 ¼ 0:

ð78Þ


The last system gives

a0 ¼ � 1

2ab;

a1 ¼1

2ab;

c ¼ �a2b;

l ¼ � 1

2aðn� 1Þ:

ð79Þ

Noting that u ¼ v1

n�1, we obtain the exact travelling wave solutions

uðx; tÞ ¼ � 1

2ab 1þ tanh

1

2aðn� 1Þ xþ a2bt

�� 1n�1

; ð80Þ

uðx; tÞ ¼ � 1

2ab 1þ coth

1

2aðn� 1Þ xþ a2bt

�� 1n�1

: ð81Þ

On the other hand, for M = 2, we set

vðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y þ a2Y 2; ð82Þto express the solution v(x, t). Substituting (82) into the reduced ODE (72) col-lecting the coefficients of Y gives the system of algebraic equations for a0, a1,

a � 2, c, and l, and solving this system we find

a0 ¼ � 1

2ab;

a1 ¼1

2ab;

a2 ¼ 0;

c ¼ �a2b;

l ¼ � 1

2aðn� 1Þ:

ð83Þ

Because a2 = 0, it is obvious that we will obtain the same solutions derived

before.

5. Generalizations of the Huxley equation

In this section we continue our goal of discussing the non-integer value of

the parameter M. It is useful to select generalizations of the Huxley equation

and the Burgers–Huxley equation to achieve our aim.


5.1. Generalization of the Huxley equation

We now consider a generalization of the Huxley equation

ut � auxx � uðk � unÞðun � 1Þ ¼ 0; n > 1: ð84Þ

This equation is used for nerve propagation in neurophysics and wall propaga-

tion in liquid crystals [1]. Eq. (84) is transformed to

�cu0 � au00 � ðk þ 1Þunþ1 þ u2nþ1 þ ku ¼ 0; ð85Þ

upon using the wave variable n = x � ct. Balancing u2n+1 with u00 gives

ð2nþ 1ÞM ¼ 4þM � 2; ð86Þ

which gives

M ¼ 1

n: ð87Þ

It is normal to use the transformation

uðx; tÞ ¼ v1n; ð88Þ

and as a result Eq. (85) becomes

�cnvv0 � anvv00 � að1� nÞðv0Þ2 � ðk þ 1Þn2v3 þ n2v4 þ kn2v2 ¼ 0: ð89Þ

Balancing v4 with vv00 gives M = 1. Consequently, we set

vðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y : ð90Þ

Substituting (90) into the reduced ODE (89), and collecting the coefficients of

Y4, Y3, Y2, Y1, and Y0 gives the following system of algebraic equations for a0,a1, c and l:

� al2a21 þ n2a41 � al2a21n ¼ 0;

� n2ka31 � n2a31 þ cnla21 þ 4n2a0a31 � 2anl2a1a0 ¼ 0;

� 3n2ka0a21 þ cnla1a0 þ 6n2a20a21 þ kn2a21 � 3n2a0a21 þ 2al2a21 ¼ 0;

4n2a30a1 þ 2kn2a0a1 � 3n2a20a1 þ 2anl2a1a0 � 3n2ka20a1 � cnla21 ¼ 0;

kn2a20 � n2ka30 � al2a21 � cnla1a0 � n2a30 þ n2a40 þ al2a21n ¼ 0:

ð91Þ


Solving this system we immediately obtain the two sets of solutions

a0 ¼1

2;

a1 ¼ � 1

2;

c ¼ �ðkðnþ 1Þ � 1Þffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r; a 6¼ 0; n > 1

l ¼ n

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðnþ 1Þ

p ;

ð92Þ

and

a0 ¼k2;

a1 ¼ � k2;

c ¼ �ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r; a 6¼ 0; n > 1;

l ¼ nk


p :

ð93Þ

This in turn gives two sets of travelling wave solutions

uðx; tÞ ¼ 1

2� 1

2tanh

n


p x� ðkðnþ 1Þ � 1Þffiffiffiffiffiffiffiffiffiffiffia

nþ 1

rt

� �" #( )1n

;

ð94Þ

uðx; tÞ ¼ 1

2� 1

2coth

n


p x� ðkðnþ 1Þ � 1Þffiffiffiffiffiffiffiffiffiffiffia

nþ 1

rt

� �" #( )1n

;

ð95Þand

uðx; tÞ ¼ k2� k2tanh

nk


p x� ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

rt

� �" #( )1n

; ð96Þ

uðx; tÞ ¼ k2� k2coth

nk


p x� ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

rt

� �" #( )1n

: ð97Þ


5.2. Generalization of the Burgers–Huxley equation

We now consider the Huxley equation

ut � auxx þ bux � uðk � unÞðun � 1Þ ¼ 0; n > 1: ð98ÞIt is obvious that for b = 0 we obtain the generalization of the Huxley equation

that was discussed before. Eq. (98) is transformed to

ðb� cÞu0 � au00 � ðk þ 1Þunþ1 þ u2nþ1 þ ku ¼ 0; ð99Þupon using the wave variable n = x � ct. Balancing u2n+1 with u00 gives

ð2nþ 1ÞM ¼ 4þM � 2; ð100Þwhich gives

M ¼ 1

n; ð101Þ

therefore we set the transformation

uðx; tÞ ¼ v1n; ð102Þ

and as a result Eq. (99) becomes

ðb� cÞnvv0 � anvv00 � að1� nÞðv0Þ2 � ðk þ 1Þn2v3 þ n2v4 þ kn2v2 ¼ 0:

ð103ÞBalancing v4 with vv00 gives M = 1. Consequently, we set

vðx; tÞ ¼ SðY Þ ¼ a0 þ a1Y : ð104ÞSubstituting (104) into the reduced ODE (103), and collecting the coefficients

of Y4, Y3, Y2, Y1, and Y0 gives the following system of algebraic equations

for a0, a1, c and l:

� al2a21 þ n2a41 � al2a21n ¼ 0;

� n2ka31 � n2a31 � nla21bþ cnla21 þ 4n2a0a31 � 2anl2a1a0 ¼ 0;

� 3n2ka0a21 þ 6n2a20a21 þ kn2a21 � 3n2a0a21 þ cnla1a0

� nla1ba0 þ 2al2a21 ¼ 0;

2kn2a0a1 þ 2anl2a1a0 þ 4n2a30a1 � 3n2ka20a1

� cnla21 þ nla21b� 3n2a20a1 ¼ 0;

kn2a20 � cnla1a0 � n2ka30 � al2a21 þ nla1ba0

� n2a30 þ n2a40 þ al2a21n ¼ 0:

ð105Þ


Solving this system we immediately obtain the two sets of solutions

a0 ¼1

2;

a1 ¼ � 1

2;

c ¼ b� ðkðnþ 1Þ � 1Þffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r; a 6¼ 0; n > 1

l ¼ n


p ;

ð106Þ

and

a0 ¼k2;

a1 ¼ � k2;

c ¼ b� ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r; a 6¼ 0; n > 1;

l ¼ nk


p :

ð107Þ

This in turn gives two sets of travelling wave solutions

uðx; tÞ ¼(1

2�1

2tanh

"n

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðnþ1Þ

p x� b�ðkðnþ1Þ�1Þffiffiffiffiffiffiffiffiffiffia

nþ1

r� �t

� �#)1n

;

ð108Þ

uðx; tÞ ¼ 1

2�1

2coth

n

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðnþ1Þ

p x� b�ðkðnþ1Þ�1Þffiffiffiffiffiffiffiffiffiffia

nþ1

r� �t

� �" #( )1n

;

ð109Þ

and

uðx; tÞ ¼ k2� k2tanh

nk


p x� b� ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r� �t

� �" #( )1n

;

ð110Þ

uðx; tÞ ¼ k2� k2coth

nk


p x� b� ðnþ 1� kÞffiffiffiffiffiffiffiffiffiffiffia

nþ 1

r� �t

� �" #( )1n

;

ð111Þ


This completes the analysis of introducing the nonlinear transformations to

overcome the situation where M becomes non-integer.

6. Discussion

The main goals of this work are to implement the standard tanh-function

method and to emphasize its power. More importantly, our purpose is to over-

come the situation where the parameter M is non-integer. It is well known that

the parameter M will be normally a positive integer so that an analytic solution

in a closed form can be derived. The cases where M is not an integer, transfor-

mation formulae should be used to overcome this complexity. We have empha-

sized in this work that this relevant transformation is powerful and can be

effectively used to discuss nonlinear evolution equations and related modelsin scientific fields.

The two goals of this work were achieved and travelling wave solutions were

formally derived to generalized forms of Burgers equation, Burgers–KdV equa-

tion, and Burgers–Huxley equation. The transformation formulae were used

for every type of nonlinearity to show that our analysis is applicable to a vari-

ety of nonlinear problems.

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Documents

Travelling wave solutions of generalized forms of Burgers, Burgers–KdV and Burgers–Huxley equations