Applied Mathematics and Computation 216 (2010) 3370–3377

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Applied Mathematics and Computation

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

Soliton solutions of Burgers equations and perturbed Burgers equation

Anwar Ja’afar Mohamad Jawad a, Marko D. Petkovic b, Anjan Biswas c,*

a Department of Electrical Engineering, University of Technology, P.O. Box 35010, Baghdad-00964, Iraqb Faculty of Science and Mathematics, Department of Mathematics and Computer Science, University of Niš, Višegradska 33, 18000 Niš, Serbiac Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications,Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA

a r t i c l e i n f o a b s t r a c t

Keywords:SolitonsExact solutionstanh method

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.04.066

* Corresponding author.E-mail address: biswas.anjan@gmail.com (A. Bisw

This paper carries out the integration of Burgers equation by the aid of tanh method. Thisleads to the complex solutions for the Burgers equation, KdV–Burgers equation, coupledBurgers equation and the generalized time-delayed Burgers equation. Finally, the per-turbed Burgers equation in (1+1) dimensions is integrated by the ansatz method.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

A number of nonlinear phenomena in many branches of sciences such as physical, chemical, economical and biologicalprocesses are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion[1–10]. One of the well known partial differential equations which govern a wide variety of them is the Burgers equation(BE) which provides the simplest nonlinear model of turbulence. The existence of relaxation time or delayed time is animportant feature in reaction diffusion and convection diffusion systems. The approximate theory of flow through a shockwave travelling is applied to viscous fluid. Fletcher, using the Hopf–Cole transformation, gave an analytic solution for thesystem of two dimensional BE.

This paper is to extend the tanh method using complex travelling waves to solve four different types of nonlinear differ-ential equations such as the Burgers, KdV–Burgers, coupled Burgers and the generalized time delayed BE.

2. Complex tanh method

Consider a general form of nonlinear partial differential equation (PDE)

Pðu;ux;ut ;uxx; . . .Þ ¼ 0: ð1Þ

Assume that the solution is given by u(x, t) = U(z) where z = i(x � kt). Hence, we use the following changes

@

@t¼ �ik

ddz;

@

@x¼ i

ddz;

@2

@x2 ¼ �d2

dz2 ;@3

@x3 ¼ �id3

dz3 ;

and so on. Eq. (1) now becomes an ordinary differential equation

QðU;U0;U00; . . .Þ ¼ 0: ð2Þ

. All rights reserved.

as).

A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377 3371

If all terms of (2) contain derivatives in z then by integrating this equation and taking the constant of integration to bezero, we obtain a simplified ODE. For the tanh method, we introduce the new independent variable

Yðx; tÞ ¼ tanhðzÞ; ð3Þ

which leads to the change of variables

ddz¼ ð1� Y2Þ d

dY;

d2

dz2 ¼ �2Yð1� Y2Þ ddYþ ð1� Y2Þ2 d2

dY2 ;

d3

dz3 ¼ 2ð1� Y2Þð3Y2 � 1Þ ddY� 6Yð1� Y2Þ2 d2

dY2 þ ð1� Y2Þ3 d3

dY3 :

ð4Þ

The next crucial step is that the solution we are looking for is expressed in the form

uðx; tÞ ¼ UðzÞ ¼Xn

i¼0

aiYi: ð5Þ

Here the parameter n can be found by balancing the highest-order linear term with the nonlinear terms in Eq. (2), Substi-tuting (5) into (2) will yield a set of algebraic equations for k, a0, a1, . . ., an since all coefficients of Y have to vanish. From theserelations, k, a0, a1, . . ., an can be obtained. Having determined these parameters and using (5) we obtain analytic solutionsu(x, t) = U(z) in a closed form. The tanh method is a powerful tool in dealing with coupled nonlinear physical models.

To illustrate the procedure, four examples related to the one dimensional Burgers, KdV–Burgers, coupled Burgers, and thegeneralized time-delayed BE are given in the following.

3. Applications

In this section, the applications of the analytical development, from the last section, will be touched upon. The BE in (1+1)dimensions will be solved. A numerical simulation will also be given. Subsequently, the KdV–BE will also be solved with thecomplex travelling wave hypothesis. Finally, the coupled BE and time-delayed BE will also be solved in the next two sub-sections. They are all supported by numerical simulations.

3.1. Burgers equation in (1+1) dimensions

Consider the one-dimensional BE which has the form [6–10]

ut þ auux � muxx ¼ 0; ð6Þ

where a and m are arbitrary constants. In order to solve Eq. (6) by the tanh method, we use the wave transformationu(x, t) = U(z), with wave complex variable z = i(x � kt). Hence, Eq. (6) becomes an ordinary differential equation:

�kiU0 þ iaUU0 þ mU00 ¼ 0: ð7Þ

Integrating Eq. (7) once with respect to z and setting the constant of integration to be 0, we obtain

�kiU þ i2aU2 þ mU0 ¼ 0: ð8Þ

By postulating tanh series, Eq. (8) reduces to

�kiU þ i2aU2 þ mð1� Y2ÞdU

dY¼ 0: ð9Þ

Now, to determine the parameter n, we balance the linear term of highest-order with the highest-order nonlinear terms.So, in Eq. (9) we balance U0 with U2 to obtain n = 1. The tanh method admits the use of the finite expansion for:

uðx; tÞ ¼ UðzÞ ¼ a0 þ a1Y :

Substituting U, U0 in Eq. (9) and equating the coefficients of Yi, i = 0, 1, 2 leads to the following nonlinear system of alge-braic equations:

� kia0 þi2aa2

0 þ ma1 ¼ 0;

� kia1 þ iaa0a1 ¼ 0;ia2

a21 � ma1 ¼ 0:

ð10Þ

Fig. 1. Solitary wave solution u(x, t) of BE for v = 0.5 and a = 0.1.

3372 A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377

Solving the nonlinear system of equations (10) we get

a0 ¼ka; a1 ¼ �

2ma

i; k ¼ 2mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a1� 2a

r; a <

12;

and therefore

uðx; tÞ ¼ 2ma

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia

1� 2a

rþ tan x� 2m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia

1� 2a

rt

!" #: ð11Þ

The solitary wave and behavior of the solution u(x, t) is new exact solution for the BE and is shown in Fig. 1 for some fixedvalues of the parameters m = 0.5, a = 0.1.

3.2. KdV–Burgers equation

Another important example is the KdV–BE [10], which can be written as

ut þ auux � muxx þ luxxx ¼ 0; ð12Þ

where a, m and l are arbitrary constants. In order to solve Eq. (12) by the tanh method, we use the wave transformationu(x, t) = U(z), with wave complex variable z = i(x + kt). Hence Eq. (12) takes the form of an ordinary differential equation:

kiU0 þ iaUU0 þ mU00 � ilU000 ¼ 0: ð13Þ

Integrating Eq. (13) once with respect to z and setting the constant of integration to be zero, we obtain:

kiU þ i2aU2 þ mU0 � ilU00 ¼ 0: ð14Þ

Using the variable transformation Y = tanh z previous equation reduces to

kU þ a2

U2 � imð1� Y2ÞdUdYþ l 2Yð1� Y2ÞdU

dY� ð1� Y2Þ2 d2U

dY2

" #¼ 0: ð15Þ

Now, to determine the parameter n, we balance the linear term of highest-order with the highest-order nonlinear terms.So, in Eq. (15) we balance U00 with U2 to obtain n = 2. Hence we assume that

uðx; tÞ ¼ UðzÞ ¼ a0 þ a1Y þ a2Y2:

Substituting U, U0, U00 in Eq. (15), then equating the coefficient of Yi, i = 0, 1, 2, 3, 4 leads to the following nonlinear systemof algebraic equations:

A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377 3373

a20a2þ a0k� ia1m� 2a2l ¼ 0;

a0a1aþ a1kþ 2a1l� 2ia2m ¼ 0;

a0a2aþa2

1a2þ ia1mþ a2kþ 8a2l ¼ 0;

a1a2a� 2a1lþ 2ia2m ¼ 0;

a22a2� 6a2l ¼ 0:

ð16Þ

Solving the nonlinear systems of equations (16) we get two classes of the solutions. First class (four solutions) is given by:

a0 ¼ �kþ 12l

a; a1 ¼ �

12m5a

i; a2 ¼12la

; k ¼ �24l; m ¼ �10il;

where each solution is obtained by choosing a different combination of signs for k and m. Second class (two solutions) is givenby:

a0 ¼ �kþ 8l

a; a1 ¼ 0; a2 ¼

12la

; k ¼ �4l; m ¼ 0:

Hence we have total of six different solitary wave solutions of (12), in the cases m = ±10il and m = 0. The solitary wave andbehavior of the solutions are new exact solutions for the BE.

3.3. Coupled Burgers equation

The third instructive example is the homogeneous form of a coupled BE [1,7]. We will consider the following system ofequations

ut � uxx þ 2uux þ aðuvÞx ¼ 0; ð17Þv t � vxx þ 2vvx þ bðuvÞx ¼ 0: ð18Þ

In order to solve Eqs. (17) and (18) by the tanh method, we use the wave transformations u(x, t) = U(z) and v(x, t) = V(z)with wave complex variable z = i(x + kt). Hence, Eqs. (17) and (18) take the form of ordinary differential equations

kiU0 þ U00 þ 2iUU0 þ aiðUVÞ0 ¼ 0; ð19ÞkiV 0 þ V 00 þ 2iVV 0 þ biðUVÞ0 ¼ 0: ð20Þ

Integrating Eqs. (19) and (20) once with respect to z and setting the constant of integration to zero, we obtain

kiU þ U0 þ iU2 þ aiUV ¼ 0; ð21ÞkiV þ V 0 þ iV2 þ biUV ¼ 0: ð22Þ

By replacing Y = tanhz, Eqs. (21) and (22) reduce to

kiU þ ð1� Y2Þ dUdYþ iU2 þ aiUV ¼ 0; ð23Þ

kiV þ ð1� Y2ÞdVdYþ iV2 þ biUV ¼ 0: ð24Þ

Balancing the order of U0 with the order of U2 and the order of V0 with the order of V2 in Eqs. (23) and (24), we find m = 1and n = 1. So the solutions take the form

uðx; tÞ ¼ UðzÞ ¼ a0 þ a1Y ; ð25Þvðx; tÞ ¼ VðzÞ ¼ b0 þ b1Y : ð26Þ

Inserting Eqs. (25) and (26) into Eqs. (23) and (24), then equating the coefficient of Yi, i = 0, 1, 2 leads to the followingnonlinear system of algebraic equations:

kia0 þ a1 þ ia20 þ iaa0b0 ¼ 0;

ka1 þ 2a0a1 þ aða0b1 þ a1b0Þ ¼ 0;

a21 þ aa1b1 ¼ 0;

kib0 þ b1 þ ib20 þ iba0b0 ¼ 0;

kb1 þ 2b0b1 þ bða0b1 þ a1b0Þ ¼ 0;

b21 þ ba1b1 ¼ 0:

ð27Þ

Fig. 2. Solitary wave solution u(x, t) of coupled BE for k = 1.2, a = 0.25, b = 2 and b0 = 1.

3374 A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377

Solving the nonlinear systems of equations (27) we can get the solution:

a0 ¼ �ðkþ ab0Þ; a1 ¼ ikða� 1Þb0; b1 ¼ ikðb� 1Þb0:

under the assumptions ab = 1, a – 1 and b – 1. Therefore we have

uðx; tÞ ¼ �ðkþ ab0Þ þ kð1� aÞb0 tanðxþ ktÞ; ð28Þvðx; tÞ ¼ b0 þ kð1� bÞb0 tanðxþ ktÞ: ð29Þ

The solutions u(x, t) and v(x, t) are shown in Figs. 2 and 3, for values of the parameters k = 1.2, a = 0.5, b = 2 and b0 = 1.

3.4. Generalized time-delayed Burgers equation

The time-delayed BE [10] is given by

sutt þ ut þ pusux � uxx ¼ 0: ð30Þ

Fig. 3. Solitary wave solution v (x, t) of coupled BE for k = 1.2, a = 0.25, b = 2 and b0 = 1.

A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377 3375

We use the wave transformation u(x, t) = U(z), with wave complex variable z = i(x � kt), Eq. (30) takes the form of an or-dinary differential equation

ð1� k2sÞU00 þ ðpUs � kÞiU0 ¼ 0: ð31Þ

Balancing U00 with UsU0 gives n = 1/s which is not an integer as s – 1. But we need the balancing number to be a positiveinteger so as to apply a transformation

U ¼ V1=s: ð32Þ

Using this transformation, Eq. (31) changes to the form:

ð1� k2sÞ 1s� 1

� �ðV 0Þ2 þ VV 00

� �þ iðpV2V 0 � kVV 0Þ ¼ 0: ð33Þ

Applying the variable change Y = tanhz, Eq. (33) reduces to

ð1� k2sÞ 1s� 1

� �1� Y2� �2 dV

dY

� �2

þ V �2Y 1� Y2� � dV

dYþ 1� Y2� �2 d2V

dY2

" #þ iðpV2 � kVÞ 1� Y2

� � dVdY¼ 0: ð34Þ

Balancing (V0)2 with V2V0 gives n = 1 and hence we may choose the following ansatz:

VðzÞ ¼ a0 þ a1Y : ð35Þ

Then

uðx; tÞ ¼ UðzÞ ¼ ½a0 þ a1Y �1=s: ð36Þ

Now, substituting (35) into (34), then equating the coefficient of Yi, i = 0, 1, 2 leads to the following nonlinear system ofalgebraic equations:

ð1� k2sÞ 1� ss

� �a1 þ ia0ðpa0 � kÞ ¼ 0;

� 2ð1� k2sÞa0 þ ia1ð2pa0 � kÞ ¼ 0;

ð1� k2sÞ 1þ ss

� �� ipa1 ¼ 0:

ð37Þ

Solving the nonlinear system of equations (37) we get

a0 ¼kð1þ sÞ

2p; a1 ¼

ið1þ sÞps

ð�1þ k2sÞ;

k ¼ � 12s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8s� s2 � ðs2 � 16sÞ1=2

2

s;

ð38Þ

and hence

uðx; tÞ ¼ 1þ sp

k2� 1

sðk2s� 1Þ tanðx� ksÞ

� �� �1=s

: ð39Þ

At the end of this section, consider the Eq. (39) in a few special cases. Let s = 4, p = 1 and s = 1. Then we obtain a0 = 5i/2,a1 = �5i/2, k = i and

uðx; tÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi52ðiþ tanðx� itÞÞ4

r: ð40Þ

Also by putting p = s = 1 and s = 0, we obtain a0 = 2i, a1 = �2i,k = 2i (by taking both plus signs in (38) and limit when s ? 0)and

uðx; tÞ ¼ 2ðiþ tanðx� 2itÞÞ: ð41Þ

4. Perturbed Burgers equation

In this section the study is going to be focused on the perturbed BE. The solitary wave ansatz method will be adopted toobtain the exact 1-soliton solution of the BE in (1+1) dimensions. The search is going to be for a topological 1-soliton solu-tion. The perturbed BE that is going to be studied in this paper is given by the following form [5]:

qt þ aqqx þ bqxx ¼ aq2qx þ bqqxx þ cðqxÞ2 þ dqxxx: ð42Þ

3376 A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377

Eq. (42) appears in the study of gas dynamics and also in free surface motion of waves in heated fluids. The perturbationterms are obtained from long-wave perturbation theory. Eq. (42) shows up in the long-wave small-amplitude limit of ex-tended systems dominated by dissipation, where dispersion is also present at a higher order [5].

In this paper the topological exact 1-soliton solution to (42) will be obtained by the solitary wave ansatz method. Thestarting hypothesis is given by [8–10]

qðx; tÞ ¼ Atanhps; ð43Þ

where

s ¼ Bðx� vtÞ: ð44Þ

Here in (43) and (44), the free parameters A and B will be determined and the unknown exponent will also be determinedduring the course of the derivation of the soliton solution to (42). Finally, the soliton velocity v will also be obtained. From(43), it is possible to obtain

qt ¼ �pvABðtanhp�1s� tanhpþ1sÞ; ð45Þqx ¼ pABðtanhp�1s� tanhpþ1sÞ; ð46Þ

qxx ¼ pðp� 1ÞAB2tanhp�2s� 2p2AB2tanhpsþ pðpþ 1ÞAB2tanhpþ2s; ð47Þ

qxxx ¼ pðp� 1Þðp� 2ÞAB3tanhp�3s� pðp� 1Þðp� 2Þ þ 2p3 AB3tanhp�1s

þ pðpþ 1Þðpþ 2Þ þ 2p3 AB3tanhpþ1s� pðpþ 1Þðpþ 2ÞAB3tanhpþ3s: ð48Þ

Substituting (45)–(48) into (42) yields

� pvABðtanhp�1s� tanhpþ1sÞ þ apA2Bðtanh2p�1s� tanh2pþ1sÞ þ bpðp� 1ÞAB2tanhp�2s� 2bp2AB2tanhps

þ bpðpþ 1ÞAB2tanhpþ2s ¼ apA3Bðtanh3p�1s� tanh3pþ1sÞ þ bpðp� 1ÞA2B2tanh2p�2s� 2bp2A2B2tanh2ps

þ bpðpþ 1ÞA2B2tanh2pþ2sþ cp2A2B2ðtanh2p�2s� 2tanh2psþ tanh2pþ2sÞ þ dpðp� 1Þðp� 2ÞAB3tanhp�3s

� dfpðp� 1Þðp� 2Þ þ 2p3gAB3tanhp�1sþ dfpðpþ 1Þðpþ 2Þ þ 2p3gAB3tanhpþ1s� dpðpþ 1Þðpþ 2ÞAB3tanhpþ3s:ð49Þ

From (49), equating exponents 2p � 1 and p yield

2p� 1 ¼ p; ð50Þ

so that

p ¼ 1: ð51Þ

It needs to be noted that the same value of p is obtained when the exponent pairs 2p + 1 & p + 2, 2p & p + 1, 2p � 2 & p � 1,2p & 3p � 1, 2p + 2 & 3p + 1 and p + 3 & 2p + 2 are equated. Now from (49), the linearly independent functions are tanh2p+jsfor j = �1, 0, 1, 2 and tanhp�1s. Thus setting their coefficients to zero yields

B ¼ a2b

A; ð52Þ

v ¼ aA2 � 2ðbþ cÞABþ 8dB2; ð53Þ2ab2 � ð2bþ cÞabþ 3da2 ¼ 0; ð54Þ

and

v ¼ 2dB2 � cAB: ð55Þ

Eq. (52) is the relation between the free parameters A and B, while the velocity of the perturbed soliton is given by (53) or(55). Finally, the constraint relation between the coefficients is given by (54) which must hold for the solitons to exist. Now,equating the two values of v also yields (54). Thus, finally the 1-soliton solution to (42) is given by

qðx; tÞ ¼ A tanh½Bðx� vtÞ�; ð56Þ

where the free parameters and the velocity are all determined above along with the constraint relations between thecoefficients.

5. Conclusions

In this paper the tanh method has been successfully applied to solve the BE in (1+1) dimensions, KdV–BE, coupled BE andthe generalized time-delayed BE. Finally, the perturbed BE in (1+1) dimensions is analyzed and solved by the ansatz method.

A.J. Mohamad Jawad et al. / Applied Mathematics and Computation 216 (2010) 3370–3377 3377

In this case, an exact topological soliton solution has been obtained together with the constraints that must be valid for thecoefficients of the equation along with the perturbation coefficients. In future, these results will be further analyzed via mul-tiple scales and He’s semi-inverse variational principle.

Acknowledgments

The second author (Marko D. Petkovic) gratefully acknowledges the support from the research project 144011 of the Ser-bian Ministry of Science.

The research work of the third author (AB) was fully supported by NSF-CREST Grant No.: HRD-0630388 and this supportis genuinely and sincerely appreciated.

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