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

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 .