Abstract and Applied Analysis

Volume 2013 (2013), Article ID 696074, 7 pages

http://dx.doi.org/10.1155/2013/696074

Research Article

Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation

Da-Quan Xian

School of Sciences, Southwest University of Science and Technology, Mianyang 621010, China

Received 21 September 2012; Accepted 8 December 2012

Academic Editor: de Dai

Copyright © 2013 Da-Quan Xian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We have undertaken the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist. The Saddle-node heteroclinic orbit has been obtained. Using the Lie group method, we get two-(1+1)-dimensional PDE, through symmetric reduction; and by the direct integral method, spread F-expansion method, and -expansion method, we obtain exact nontraveling wave solutions, for the (2+1)D KdV Burgers equation, and find out some new strange phenomenons of sympathetic vibration to evolution of nontraveling wave.

1. Introduction

We consider the (2+1)-dimensional Korteweg-de Vries Burgers ((2+1)D KdV Burgers) equation

where , , , and are real parameters. Equation (1) is model equation for wide class of nonlinear wave models in an elastic tube, liquid with small bubbles, and turbulence [1–3]. Much attention has been put on the study of their exact solutions by some methods [4], such as, a complex line soliton by extended tanh method with symbolic computation [5], exact traveling wave solutions including solitary wave solutions, periodic wave and shock wave solutions by extended mapping method, and homotopy perturbation method [6, 7].

It is well known that the investigation of exact solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. Many effective methods have been presented [7–22], such as functional variable separation method [8, 9], homotopy perturbation method [12], F-expansion method [7, 13], Lie group method [14, 15], variational iteration method [16], homoclinic test method [17–19], Exp-function method [20, 21], and homogeneous balance method [22]. Practically, there is no unified method that can be used to handle all types of nonlinearity.

In this paper, we will discuss the existence of periodic traveling wave solution and seek the Saddle-Node heteroclinic orbit, and further use the Lie group method with the aid of the symbolic computation system Maple to construct the non-traveling wave solutions for (1).

2. Existence of Periodic Traveling Wave Solution of (1)

Introducing traveling wave transformation in this form

permits us to convert (1) into an ODE for

where , Integrating (3) with respect to twice and taking integration constant to yields

Letting , thus nonlinear ordinary differential equation (4) is equivalent to the autonomous dynamic system as follows:

The dynamic system (5) has two balance points:

The Jacobi matrixes at the balance points for the right-hand side of (5) are obtained as follows, respectively:

Their latent equations are expressed, respectively, as,

Relevant latent roots are as follows respectively:

Obviously, if , then are two positive real roots, therefore is a nonsteady node point. If , then are conjugate complex roots and real part is positive, so is a nonsteady focus point. And is a positive and minus real root, thus is a saddle point. From (5), we know the phase trajectory on the phase plane satisfies

Integrating (11), we can obtain

where is a total energy or Hamiliton function of system (4). Apparently

Consequently, the system expressed in (12) is not a conservative one, then periodic traveling wave solution of (1) does not exist.

We conclude the above analysis in the following theorem.

Theorem 1. Under the traveling wave transformation, the periodic solution of (2+1)-dimensional KdV-Burgers equation does not exist.

But, saddle-node heteroclinic orbits and nontraveling periodic solution do exist, which will be discussed later in this paper.

3. Saddle-Node Heteroclinic Orbits of KdV-Burgers Equation

First, we assume the solutions of (4) in the form

Substituting (14) into (4) yields

Then we get

Solving the system (16) gets

Substituting (17) into (14) obtains

Evidently, , . Thus (18) is a saddle-node heteroclinic orbit through nonsteady node point and saddle point [23].

Ecumenic, taking the Hamiliton function , we obtain

where is an arbitrary constant. Integrating (19) with respect to we have

where is an arbitrary constant. We can see that (4) has the general solution (20) and all partial cases as include above result can be found from the general solution of (20). Example, take , , in (20), we find a solution of (4) as follows:

It is a heteroclinic orbit too.

4. Li Symmetry of (1)

This section devotes to Li symmetry of (1) [14, 15]. Let

be the Li symmetry of (1). From Lie group theory, satisfies the following equation

We take the function in the form

where   () are functions to be determined later. Substituting (3) into (2) yields

where   () are arbitrary functions of , is an arbitrary constant. Substituting (25) into (24), we obtain the Li symmetries of (1) as follows:

5. Symmetry Reduction and Solutions of (1)

Based on the integrability of reduced equation of symmetry (26), we are to consider the following three cases.

Case 1. Taking and in (26) yields

The solution of the differential equation is

Substituting (28) into (1) yields the function which satisfies the following linear PDE:

By integrating both sides, we find out the following result:

where , are new arbitrary functions of . Substituting (30) into (28), we can get the solutions of (1) as follows:

(1) Given   (), , in (31), the local structure of is obtained (Figure 1). Where is an Jacobian elliptic cosine function.

(2) Given , , , , in (31), the local structure of is obtained (Figure 2).

Figure 1: The strange phenomenon which is a sympathetic vibration of periodicity on the -axis and paraboloid on -axis for as .

Figure 2: The periodic solution which is a periodic nontraveling wave traveling on the -axis for as .

Case 2. Take , and in (26), then

Solving the differential equation , we can get

Substituting (33) into (1) and integrating once with respect to yield

Again, further using the transformation of dependent variable to (34),

Substituting (35) into (34) and integrating once with respect to yield

where is an integration constant, . We assume that the solution of (36) can be expressed in the form

where   () are constants to be determined later, satisfies the following auxiliary equation

Substituting (37) and (38) into (36) and equating the coefficients of all powers of to zero yield a set of algebra equations for , , , and as follows.

Solving the system of function equations with the aid of Maple, we obtain

when , , , where .

It is known that solutions of (38) are as follows [24]:

Substituting (41), (40), (37), and (35) into (33), we obtain solutions of (1) as follows:

(see Figures 3 and 4).

Figure 3: Local structure of is shown as ,  ,  ,  ,  ,  and .

Figure 4: Local structure of is shown as ,  ,  ,  ,  ,  .

Remark 2. If we direct assume that the solution of (34) can be expressed in the form

where , , and are continuous functions of to be determined later. satisfies the auxiliary equation (38). Substituting (43) and (38) into (34), equating the coefficients of all powers of to zero yields a set of function equations for , , , , and as follows:

Solving the system of function equations, we obtain

This result indicate the idea is equivalent to idea of Case 2 above.

Case 3. Take and in (26), then

Solving the differential equation , we obtain

Substituting (47) into (1) yield

Using the transformation and integrating the resulting equation with respect to we have

where is an arbitrary constant, . Suppose that the solution of ODE (49) can be expressed by a polynomial in as follows:

where satisfies the second-order LODE in the form [25]

Balancing with in (49) gives . So that

where   () and are constants to be determined later. Substituting (52) and (51) into (49). Setting these coefficients of the to zero, yields a set of algebraic equations as follows:

Solving the algebraic equations above yields

when and . Consequently, we obtain the following solution of (1) for :

where .

6. Conclusions

Based on the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist, we have obtained Saddle-node Heteroclinic Orbits. By applying the Lie group method, we reduce the (2+1)D KdV Burgers equation to (1+1)-dimensional equations including the (1+1)-dimensional linear partial differential equation with constants coefficients (29), (48) and (1+1)-dimensional nonlinear partial differential equation with variable coefficients (34). By solving the equations (29), (34), and (48), we obtain some new exact solutions and discover the strange phenomenon of sympathetic vibration to evolution of nontraveling wave soliton for the (2+1)D KdV Burgers equation. Our results show that the unite of Lie group method with others is effective to search simultaneously exact solutions for nonlinear evolution equations. Other structures of solutions with symmetry (26) are to be further studied.

Acknowledgments

The authors would like to thank professor S. Y. Lou for the helpful discussions. This work was supported by key research projects of Sichuan Provincial Educational Administration no. 10ZA021 and Chinese Natural Science Foundation Grant no. 10971169.

References

R. S. Johnson, “A non-linear equation incorporating damping and dispersion,” Journal of Fluid Mechanics, vol. 42, pp. 49–60, 1970.

L. van Wijngaarden, “One-dimensional flow of liquids containing small gas bubbles,” Annual Review of Fluid Mechanics, vol. 4, pp. 369–396, 1972.

G. Gao, “A theory of interaction between dissipation and dispersion of turbulence,” Scientia Sinica A, vol. 28, no. 6, pp. 616–627, 1985.

W. X. Ma, “An exact solution to two-dimensional Korteweg-de Vries-Burgers equation,” Journal of Physics A, vol. 26, no. 1, pp. L17–L20, 1993.

Z. B. Li and M. L. Wang, “Travelling wave solutions to the two-dimensional KdV-Burgers equation,” Journal of Physics A, vol. 26, no. 21, pp. 6027–6031, 1993.

J. Zhang, “Soliton-like solutions for the -dimensional nonlinear evolution equation,” Communications in Theoretical Physics, vol. 32, no. 2, pp. 315–318, 1999.

E. G. Fan and H. Q. Zhang, “A note on the homogeneous balance method,” Physics Letters A, vol. 246, pp. 403–406, 1998.

S. Y. Lou and G. J. Ni, “The relations among a special type of solutions in some -dimensional nonlinear equations,” Journal of Mathematical Physics, vol. 30, no. 7, pp. 1614–1620, 1989.

X.-Y. Tang and S.-Y. Lou, “Folded solitary waves and foldons in dimensions,” Communications in Theoretical Physics, vol. 40, no. 1, pp. 62–66, 2003.

E. J. Parkes and B. R. Duffy, “Travelling solitary wave solutions to a compound KdV-Burgers equation,” Physics Letters A, vol. 229, no. 4, pp. 217–220, 1997.

A. Elgarayhi and A. Elhanbaly, “New exact traveling wave solutions for the two-dimensional KdV-Burgers and Boussinesq equations,” Physics Letters A, vol. 343, no. 1–3, pp. 85–89, 2005.

A. Molabahrami, F. Khani, and S. Hamedi-Nezhad, “Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method,” Physics Letters A, vol. 370, no. 5-6, pp. 433–436, 2007.

H. L. Chen and D. Q. Xian, “Periodic wave solutions for the Klein-Gordon-Zakharov equations,” Acta Mathematicae Applicatae Sinica. Yingyong Shuxue Xuebao, vol. 29, no. 6, pp. 1139–1144, 2006.

D. Q. Xian, “New exact solutions to a class of nonlinear wave equations,” Journal of University of Electronic Science and Technology of China, vol. 35, no. 6, pp. 977–980, 2006.

D.-Q. Xian and H.-L. Chen, “Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1340–1349, 2010.

J.-H. He, “Periodic solutions and bifurcations of delay-differential equations,” Physics Letters A, vol. 347, no. 4–6, pp. 228–230, 2005.

Z. Dai and D. Xian, “Homoclinic breather-wave solutions for Sine-Gordon equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 3292–3295, 2009.

Z. Dai, D. Xian, and D. Li, “Homoclinic breather-wave with convective effect for the - dimensional boussinesq equation,” Chinese Physics Letters, vol. 26, no. 4, Article ID 040203, 2009.

Z. Dai, Z. Li, Z. Liu, and D. Li, “Exact homoclinic wave and soliton solutions for the 2D Ginzburg-Landau equation,” Physics Letters A, vol. 372, no. 17, pp. 3010–3014, 2008.

D. Xian and Z. Dai, “Application of exp-function method to potential kadomtsev-petviashvili equation,” Chaos, Solitons and Fractals, vol. 42, pp. 2653–2659, 2009.

G.-C. Xiao, D.-Q. Xian, and X.-Q. Liu, “Application of exp-function method to Dullin-Gottwald-Holm equation,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 536–541, 2009.

X.-Q. Liu and S. Jiang, “The --method and its applications,” Physics Letters A, vol. 298, no. 4, pp. 253–258, 2002.

S. K. Liu and S. D. Liu, Solitary wave and turbulence, Shanghai Scientifc and Technological Education, Shanghai, China, 1994.

S. K. Liu and S. D. Liu, Nonliner Equations in Physics, Peking University, Beijing, China, 2000.

M. Wang, X. Li, and J. Zhang, “The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.