exact travelling wave solutions for the dissipative (2 + 1)-dimensional akns equation

Applied Mathematics and Computation 217 (2010) 735–744

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Applied Mathematics and Computation

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

Exact travelling wave solutions for the dissipative (2 + 1)-dimensionalAKNS equation

Liu Qiang a,b,*, Zhang Weiguo a

a School of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR Chinab Department of Mathematics and Information Science, The College of Zhengzhou Light Industry, Zhengzhou, Henan 450002, PR China

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

Keywords:The dissipative (2 + 1)-dimensional AKNSequationGlobal phase portraitBell profile solitary wave solutionKink profile solitary wave solutionsDamped oscillatory solutionError estimates

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.06.011

* Corresponding author at: Department of MathePR China.

E-mail address: [email protected] (L. Qiang)

This paper employs the theory of planar dynamical systems and undetermined coefficientmethod to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNSequation. By qualitative analysis, global phase portraits of the dynamic system correspond-ing to the equation are obtained under different parameter conditions. Furthermore, therelations between the properties of travelling wave solutions and the dissipation coeffi-cient r of the equation are investigated. In addition, the possible bell profile solitary wavesolution, kink profile solitary wave solutions and approximate damped oscillatory solu-tions of the equation are obtained by using undetermined coefficient method. Error esti-mates indicate that the approximate solutions are meaningful. Based on above studies, amain contribution in this paper is to reveal the dissipation effect on travelling wave solu-tions of the dissipative (2 + 1)-dimensional AKNS equation.

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1. Introduction

Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of sci-ence, especially in physics. In 1972, AKNS [1] reduced the KdV equation to the (2 + 1)-dimensional AKNS equation by use ofthe inverse Scattering transform (IST). The (2 + 1)-dimensional AKNS equation is

ut � uxxt � 4uut � 2ux@�1y ut þ ux ¼ 0: ð1:1Þ

At the same time they also pointed out that the (2 + 1)-dimensional AKNS equation has the desirable properties of the PBBM[2,3] equation in that it responds feebly to short waves. It has of course the additional property that it is exactly solvable.

Some solutions of Eq. (1.1) have been obtained by using Hirota bilinear method, tanh–coth method, Exp-function method,Darboux transformation method and multi-linear variable separation approach [4–8]. In additional, Lü et al. [9] have studiedthe (2 + 1)-dimensional AKNS equation with variable coefficients and have got some new explict solutions by applying Liesymmetry method.

As the shallow water wave will inevitably lead to dissipation when it encounter damping, Whitham [10] indicated thatone fundamental question of the nonlinear evolution equations would be how about the effect of viscous amendment. There-fore, we study the dissipative (2 + 1)-dimensional AKNS equation

ut � uxxt � 4uut � 2ux@�1y ut þ ux � ruxt ¼ 0; ð1:2Þ

where r is constant. r – 0 shows that the system has dissipative effect.

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matics and Information Science, The College of Zhengzhou Light Industry, Zhengzhou, Henan 450002,

.

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736 L. Qiang, Z. Weiguo / Applied Mathematics and Computation 217 (2010) 735–744

In this article we consider the existence of the travelling wave solutions of Eq. (1.2). This paper is organized as follows. InSection 2, we carry out qualitative analysis for the dynamic system corresponding to Eq. (1.2), which will help us to fullyunderstand the number and behavior of the bounded traveling wave solutions, as well as the evolving relationship betweenthe rail line. In Section 3, we study the effect of dissipation on the travelling wave solutions of Eq. (1.2). In Section 4, thepossible bell profile solitary wave solution and kink profile solitary wave solutions of Eq. (1.2) are presented. Furthermore,based on the cause and structure of damped oscillatory solutions, approximate damped oscillatory solutions are obtainedunder various conditions. In Section 5, we give error estimates for the approximate damped oscillatory solutions gottenin Section 4. Through the investigation in this paper, we can comprehend the influence of dissipation term on travelling wavesolutions of Eq. (1.2).

2. Existence of bell profile solitary wave solution, kink profile solitary wave solutions and oscillatory travelling wavesolutions of Eq. (1.2)

Suppose that u(x,y, t) = u(n) = u(x + by � vt) is a travelling wave solution of Eq. (1.2). Substituting it into Eq. (1.2) leads to

�vu0ðnÞ þ vu000ðnÞ þ 4vuðnÞu0ðnÞ þ 2vb

uðnÞu0ðnÞ þ u0ðnÞ þ rvu00ðnÞ ¼ 0: ð2:1Þ

Integrating Eq. (2.1) once with respect to n yields (2.1)

u00ðnÞ þ ru0ðnÞ þ 2þ 1b

� �u2ðnÞ þ 1

v � 1� �

uðnÞ þ c ¼ 0; ð2:2Þ

where c is an integral constant. Thus, to study the existence of bounded travelling wave solutions of Eq. (1.2) is equivalent tostudy the existence of bounded solutions of Eq. (2.2). To this end, let x = u(n), y = u0(n), then Eq. (2.2) can be reformulated as aplanar dynamic system

dxdn ¼ y , Pðx; yÞ;dydn ¼ �ry� 2þ 1

b

� �x2 � 1

v � 1� �

x� c , Qðx; yÞ:

8<: ð2:3Þ

Owing to @P@x þ

@Q@y ¼ �r, by Bendixson–Dulac’s criterion [12], we have the following proposition for system (2.3).

Proposition 2.1. If r – 0, then system (2.3) does not have any closed orbit or singular closed orbit with finite number of singularpoints on (x,y) phase plane. Further, there exists no periodic travelling wave solution or bell profile solitary wave solution of Eq.(1.2) as r – 0.

In the (x,y) plane, the number of bounded singularities in system (2.3) depends on the number of solutions in equation of

f ðxÞ ¼ 2þ 1b

� �x2 þ 1

v � 1� �

xþ c ¼ 0: ð2:4Þ

For clarity and non-repetitiveness, we assume that 4 2þ 1b

� �c < 0 throughout the paper. Other cases can be discussed sim-

ilarly. Obviously, system (2.3) has two singular points Pi(xi,0)(i = 1,2), where x1 ¼� 1

v�1ð Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v�1ð Þ2�4c 2þ1

bð Þq

2 2þ1bð Þ and x2 ¼

� 1v�1ð Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v�1ð Þ2�4c 2þ1

bð Þq

2 2þ1bð Þ :

We use

Jðxi;0Þ ¼0 1

�f 0ðxiÞ �r

� �i ¼ 1;2;

to denote the Jacobi matrix of the linearized system of system (2.3) at singular points Pi(xi,0)(i = 1,2).

2.1. Finite singular points of the system (2.3)

1. In the case of r = 0In this case, system (2.3) has first integral

Hðx; yÞ ¼ 12

y2 þ 13

2þ 1b

� �x3 þ 1

21v � 1� �

x2 þ cx ¼ h: ð2:5Þ

It is easy to see that P2 is a center, P1 is a saddle point.

L. Qiang, Z. Weiguo / Applied Mathematics and Computation 217 (2010) 735–744 737

2. In the case of r > 0The discriminants corresponding to these characteristic equations are denoted by:

Mi ¼ r2 � 8 2þ 1b

� �x2 þ 4

1v � 1� ��

; i ¼ 1;2:

(1) If D2 > 0, namely r2 � 8 2þ 1b

� �x2 þ 4 1

v � 1� ��

> 0; P2 is stable node point;

(2) If D2 < 0, namely r2 � 8 2þ 1b

� �x2 þ 4 1

v � 1� ��

< 0; P2 is stable focus point.3. In the case of r < 0

(1) If D2 > 0, namely r2 � 8 2þ 1b

� �x2 þ 4 1

v � 1� ��

> 0; P2 is unstable node point;

(2) If D2 < 0, namely r2 � 8 2þ 1b

� �x2 þ 4 1

v � 1� ��

< 0; P2 is unstable focus point.

2.2. Infinite singular points of the system (2.3)

Applying Poincaré transformation to analyze singular points at infinity of system (2.3), it is clear that there exist a coupleof singular points at infinity E1, E2 on y axis (E1 is source point and E2 is Meeting Point), meanwhile, the circumference ofPoincaré disk is orbits.

Theorem 2.1.

(1) If r = 0, then Eq. (1.2) has bell profile solitary wave solution;(2) If r – 0, then Eq. (1.2) has kink profile solitary wave solutions and oscillatory travelling wave solutions.

Based on the above results and the theory of planar dynamical systems, we present the global phase portraits of system(2.3) (see Figs. 1–5).

3. Relations between the properties of bounded travelling wave solutions and the parameter r of Eq. (1.2)

Under the qualitative analysis in last section, we obtained the existence of bounded traveling wave solutions in Eq. (1.2).In this section, we will study the relationship between the behavior of the bounded traveling wave solutions and the dissi-pative coefficient r in Eq. (1.2), and obtain the necessary and sufficient conditions that bounded travelling wave solution ap-pear as kink profile solitary wave solution or Non-monotonic traveling wave solution (including damped oscillatorysolution).

Let r21 ¼ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2 � 4c 2þ 1

b

� �r, we can arrive at

Fig. 1. r = 0.

Fig. 2. r > 0, D2 P 0.

Fig. 3. r > 0, D2 < 0.


Theorem 3.1

(1) If r < �jr1j, Eq. (1.2) has monotone decreasing kink profile solution u2(n) satisfies the case

u2ð�1Þ ¼� 1

v � 1� �

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; u2ðþ1Þ ¼� 1

v � 1� �

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � :

(corresponding to L(P2,P1) shown in Fig. 4.)(2) If

r > jr1j, Eq. (1.2) has monotone increasing kink profile solution u2(n), satisfying the case

u2ð�1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; u2ðþ1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � :

(corresponding to L(P1,P2) shown in Fig. 2.)

Fig. 4. r < 0, D2 P 0.

Fig. 5. r < 0, D2 < 0.


Theorem 3.2

(1) If �jr1j < r < 0, then Eq. (1.2) has an oscillatory solution u3(n), satisfying the case

u3ð�1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; u3ðþ1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � :

(corresponding to L(P2,P1) shown in Fig. 5.) has maximum at �n1. The solution has the property of monotonically increasing at theright of �n1 and damped at the left of �n1, that is, there exist numerably infinite maximum points �niði ¼ 1;2; . . .Þ and minimum pointsni(i = 1,2, . . .) on n axis, such that

�1 < � � � < nn < �nn < � � � < n1 < �n1 < þ1;limn!1

�nn ¼ limn!1

nn ¼ þ1;

(ð3:1Þ

and


u3ðþ1Þ < u3ðn1Þ < u3ðn2Þ < � � � < u3ðnnÞ < � � � < u3ð�1Þ < � � � < u3ð�nnÞ < u3ð�nn�1Þ < � � � < u3ð�n1Þ;

limn!1

u3ð�nnÞ ¼ limn!1

u3ðnnÞ ¼ u3ð�1Þ:

8<:limn!1ð�nn � �nnþ1Þ ¼ lim

n!1ðnn � nnþ1Þ ¼ 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v�1ð Þ2�4c 2þ1

bð Þq

�r2

r :ð3:2Þ

(2) If 0 < r < jr1j, Eq. (1.2) has an oscillatory solution u4(n), satisfying the case

Vð�1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; Vðþ1Þ ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � :

(corresponding to L(P1,P2) shown in Fig. 3 .) has maximum at �n1, The solution has the property of monotonically increasing at theleft of �n1 and damped at the right of �n1, that is, there exist numerably infinite maximum points �niði ¼ 1;2; . . .Þ on n and minimumpoints ni(i = 1,2, . . .) on n axis, such that

�1 < �n1 < n1 < � � � < �nn < nn < � � � < þ1;

limn!1

�nn ¼ limn!1

nn ¼ þ1;

8<: ð3:3Þ

and

u4ð�1Þ < u4ðn1Þ < u4ðn2Þ < � � � < u4ðnnÞ < � � � < u4ðþ1Þ < � � � < uð�nnÞ < u4ð�nn�1Þ < � � � < u4ð�n1Þ;limn!1

u4ð�nnÞ ¼ limn!1

u4ðnnÞ ¼ u4ðþ1Þ;

(

limn!1ð�nn � �nnþ1Þ ¼ lim

n!1ðnn � nnþ1Þ ¼ 4pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1v�1Þ2�4cð2þ1

bÞp

�r2

q :ð3:4Þ

From Theorems 3.1 and 3.2, we know that the dissipation term ruxt enables Eq. (1.2) to have kink profile solitary wavesolutions or oscillatory travelling wave solutions. When jrj is large, Eq. (1.2) has at least a monotone travelling wave solution,which appears as a kink profile solitary wave solution; when jrj is small, Eq. (1.2) has oscillatory travelling wave solution,which has damped property (it is called damped oscillatory solution in this paper).

4. Bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatorysolutions of Eq. (1.2)

4.1. Bell profile solitary wave solution and kink profile solitary wave solutions of Eq. (1.2)

By the analysis in the second section and the third section, we know that the bell profile solitary wave solution of Eq. (1.2)exist when r equal to zero, and if r is large enough, we can obtain kink profile solitary wave solutions of the Eq. (1.2).

From [11], we can obtain bell profile solitary wave solution u1(n) (see Fig. 6) corresponding to L(P1,P1) shown in Fig. 1.

Fig. 6. The graphics of u1(n) with b = 0.1, v = 1.5, c = �0.5, n0 = 0.3.


u1ðnÞ ¼3

2ð2þ 1bÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2

� 4c 2þ 1b

� �s� sec h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2 � 4cð2þ 1

bÞ4q

2ðn� n0Þ

0@

1A

�1v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � : ð4:1Þ

At the same time, u(n) has kink profile solitary wave solutions of the form

u2ðnÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2 � 4cð2þ 1

bÞqð2þ 1

bÞ½1þ erðn�n0Þ�2�

1v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � : ð4:2Þ

If r > 0, u2(n) (see Fig. 7) corresponds to L(P1,P2) shown in Fig. 2, meanwhile, if r < 0, u2(n) (see Fig. 8) corresponds to L(P2,P1)shown in Fig. 4.

4.2. Approximate damped oscillatory solutions of Eq. (1.2)

This section studies approximate damped oscillatory solutions of Eq. (1.2). Note that the discriminant of the characteristicequation of Eq. (2.3) at singular points. Analysis by the last section, we know that damped oscillatory solutions of Eq. (1.2)will arise when jrj is small. Since its accurate expressions are difficult to obtain, the following, we will give the approximatesolutions of the damped oscillatory solutions. It is easy to see that saddle-focus orbits L(P1,P2) in Fig. 3 come from the breakof homoclinic orbit L(P1,P1) in Fig. 1 under the effect of dissipation term ruxt. Hence, the non-oscillatory part of the dampedoscillatory solution corresponding to L(P1,P2) can be denoted by the bell profile solitary wave solution of the form

Fig. 7. The graphics of u2(n) with a = 2, b = 0.1, v = 1.5, c = �0.5, n0 = 0.3, r = 1.

Fig. 8. The graphics of u2(n) with a = 2, b = 0.1, v = 1.5, c = �0.5, n0 = 0.3, r = �1.


u�ðnÞ ¼ 3

2 2þ 1b

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v � 1� �2

� 4c 2þ 1b

� �s� sec h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1v � 1Þ2 � 4cð2þ 1

bÞ4q

2ðn� n0Þ

0@

1A

�1v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; ð4:3Þ

which is gotten in (4.1), while the oscillatory part of the damped oscillatory wave solution can be expressed approximativelyby the following form

uðnÞ ¼ eaðn�n0ÞðA1 cosðBðn� n0ÞÞ � A2 sinðBðn� n0ÞÞÞ þ C; ð4:4Þ

where A1, A2, B, C, a are undetermined constants.Substituting (4.4) into Eq. (2.2), and neglecting the terms including e2aðn�n0Þ, we have

B2 ¼ 1v � 1þ raþ a2 þ 4C þ 2C

b ;

Bðr þ 2aÞ ¼ 0;

C ¼� 1

v�1ð Þþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v�1ð Þ2�4c 2þ1

bð Þq

2 2þ1bð Þ :

8>>>><>>>>:

ð4:5Þ

That is

B2 ¼


� 4c 2þ 1b

� �s� r2

4: ð4:6Þ

The value of A1cos(B(n � n0)) � A2sin(B(n � n0))) + C is independent of the value of B, without lose of generality, let B > 0throughout this paper.

In order to derive approximate damped oscillatory solution of Eq. (1.2), there still require some conditions to connect(4.3) and (4.4). We choose

di

dniuðn0Þ ¼

di

dniu�ðn0Þ; i ¼ 0;1;

namely, we take

A1 þ C ¼ u�ðn0Þ; aA1 � A2B ¼ 0; ð4:7Þ

as connective conditions. Then we can obtain

A1 ¼


b

� �r

2 2þ 1b

� � ; A2 ¼rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2 � 4c 2þ 1

b

� �r� r2

4

s : ð4:8Þ

According to the analysis above, we can arrive at:

Theorem 4.1

(1) When �jr1j < r < 0, Eq. (1.2) has a damped oscillatory solution u3(n) (see Fig. 9) corresponding to focus-saddle orbit L(P2,P1)in Fig. 5, whose approximate expression is

u3ðnÞ �e�

r2ðn�n0ÞðA1 cosðBðn� n0ÞÞ � A2 sinðBðn� n0ÞÞÞ þ C; n 2 ð�1; n0�;

3A sec h2 a2 ðn� n0Þ� �

þ D; n 2 ðn0;þ1Þ;

(ð4:9Þ

where

A ¼


b

� �r

2 2þ 1b

� � ; a ¼


� 4c 2þ 1b

� �4

s;

C ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; D ¼� 1

v � 1� �


1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ;

Fig. 9. The graphics of u3(n) with b = 0.1, v = 1.5, c = �0.5, n0 = 0.3, r = �0.2.

Fig. 10. The graphics of u4(n) with b = 0.1, v = 1.5, c = �0.5, n0 = 0.3, r = 0.2.


where A1, A2, B are given from (4.8) and (4.6);(2) When 0 < r < jr1j, Eq. (1.2) has a damped oscillatory solution u4(n) (see Fig. 10) corresponding to saddle-focus orbit L(P1,P2)

in Fig. 3, whose approximate expression is

u4ðnÞ �3A sec h2 a

2 ðn� n0Þ� �

þ D; n 2 ð�1; n0�;e�

a2ðn�n0ÞðA1 cosðBðn� n0ÞÞ � A2 sinðBðn� n0ÞÞÞ þ C; n 2 ðn0;þ1Þ;

(ð4:10Þ

where A, a, D, A1, A2, B, C are ditto.

5. Error estimates of approximate damped oscillatory solutions of Eq. (1.2)

In this section, we investigate error estimates between exact damped oscillatory solutions and the approximate solutionsgiven in Section 4. We still take (4.10) as an example. Error estimates between other damped oscillatory solutions and theirapproximate solutions can be obtained similarly. In order to obtain error estimate of the approximate damped oscillatorysolution corresponding to L(P1,P2) shown in Fig. 3, we substitute

VðnÞ ¼ uðnÞ � x2

x1 � x2; ð5:1Þ

into Eq. (2.2). Consequently,

V 00ðnÞ þ rV 0ðnÞ � 2 2þ 1b

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1v � 1� �2

� 4c 2þ 1b

� �sVðnÞðVðnÞ � 1Þ ¼ 0: ð5:2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

Since r2 < 4 1v � 1� �2 � 4c 2þ 1

b

� �, it is easy to obtain the inexplicit expression of the general solution by constant variation

method


uðnÞ � x2 ¼ eaðn�n0ÞðA1 cosðBðn� n0ÞÞ � A2 sinðBðn� n0ÞÞÞ þ2þ 1

b

B

Z n

n0

eaðn�n0�sÞ sinðBðn� n0 � sÞÞðuðsÞ � x2Þ2ds; ð5:3Þ

which satisfy

uðn0Þ ¼� 1

v � 1� �

þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1v � 1� �2 � 4c 2þ 1

b

� �r

2 2þ 1b

� � ; u0ðn0Þ ¼ 0;

where a ¼ � r2 ;A1;A2 and B are given from (4.8) and (4.6). Thus (5.3) reflects the relation between approximate damped oscil-

latory solution (4.10) and the inexplicit expression of the general solution.Owing to the damped oscillatory solution is bounded, there exists K1 > 0 such that ju(n)j < K1. Furthermore, from (5.3), we

have

juðnÞ � x2j 6 C1e�r2ðn�n0Þ þ T1ð1þ 2bÞ

Bb

Z n

n0

e�r2ðn�n0�sÞjuðnÞ � x2jds; n > n0;

where C1 = jA1j + jA2j, T1 = K1 + jx2j. By Gronwall inequality, the above formula becomes

juðnÞ � x2j 6 C2e�r2ðn�n0Þ; n > n0; ð5:4Þ

where C2 ¼ C1e�T1 ð1þ2bÞ

rBb . (5.4) is the amplitude estimate of damped oscillatory solution of Eq. (1.2). From (5.4), it is obviouslythat n is approaching to x2 when n is approaching to +1.

Further, by (5.4) and (5.3) available, we have

uðnÞ � e�r2ðn�n0ÞðA1 cosðBðn� n0ÞÞ � A2 sinðBðn� n0ÞÞÞ þ x2

� � 6 � rBb1þ 2b

e�rðn�n0Þ 1� e�r2ðn�n0Þ

� �; n > n0: ð5:5Þ

Eqs. (5.4) and (5.5) show that the error estimate between the exact damped oscillatory solution and the approximate solu-tion (4.10) is less than

e1ðnÞ ¼ �rBb

1þ 2be�rðn�n0Þð1� e�

r2ðn�n0ÞÞ:

Due to e1ðnÞ ¼ oðe�rðn�n0ÞÞ as n ? +1. (4.10) is meaningful to be an approximate solution of Eq. (1.2) when the conditions inTheorem 4.1(i) hold. In the same way, (4.9) is meaningful to be an approximate solution of Eq. (1.2) when the conditions inTheorem 4.1(ii) hold.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 10701066), the Shanghai Natural Sci-ence Foundation (No. 10ZR1420800), Shanghai Leading Academic Discipline Project (No. S30501) and the Innovation FundProject for Graduate Student of Shanghai (No. JWCXSL0901).

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exact travelling wave solutions for the dissipative (2 + 1)-dimensional akns equation

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