Abstract and Applied Analysis

Volume 2012 (2012), Article ID 109235, 14 pages

http://dx.doi.org/10.1155/2012/109235

## Kink Waves and Their Evolution of the RLW-Burgers Equation

^{1}School of Mathematics, Chengdu University of Information Technology, Sichuan, Chengdu 610225, China^{2}School of Computer Science Technology, Southwest University for Nationalities, Sichuan, Chengdu 610041, China

Received 13 March 2012; Revised 26 May 2012; Accepted 11 June 2012

Academic Editor: Victor M. Perez Garcia

Copyright © 2012 Yuqian Zhou and Qian Liu. 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

This paper considers the bounded travelling waves of the RLW-Burgers equation. We prove that there only exist two types of bounded travelling waves, the monotone kink waves and the oscillatory kink waves. For the oscillatory kink wave, the regularity of its maximum oscillation amplitude changing with parameters is discussed. Exact expressions of the monotone kink waves and approximate expressions of the oscillatory ones are obtained in some special cases. Furthermore, all bounded travelling waves of the RLW-Burgers equation under different parameter conditions are identified and the evolution of them is discussed to explain the corresponding physical phenomena.

#### 1. Introduction

The RLW-Burgers equation,

In 1981, Bona et al. [1] developed a numerical scheme to solve (1.1) and found that the model could give quite a good description of the spatial and temporal development of periodically generated waves. In 1989 Amick et al. [3] discussed large-time behavior of solutions to the initial-value problem of (1.1) and used the methods such as energy estimates, a maximum principle, and a transformation of Cole-Hopf type to obtain sharp rates of temporal decay of certain norms of the solution. Later, travelling wave solutions of (1.1) were considered due to their important roles in understanding the complicated nonlinear wave phenomena and long-time behavior of solution. People paid more attention to some special exact travelling wave solutions of (1.1) because of the nonintegrability of travelling wave system of it. In [4], Zhang and Wang gave an exact solution of (1.1) for

Though there have been some profound results about travelling wave solutions of (1.1) which contributed to our understanding of nonlinear physical phenomena and wave propagation, there still exist some unresolved problems from the viewpoint of physics. For instance, are there other types of bounded travelling waves such as solitary waves, periodic waves, and oscillatory travelling waves? If they exist, how do they evolve? How does the oscillatory amplitude of the oscillatory travelling waves vary with dissipative and dispersive parameters? How can we get their exact expressions and plot their wave profiles? To answer these questions, we need to figure out how the travelling wave solutions of (1.1) depending on the parameters. In fact, it has involved bifurcation of travelling wave solutions. In general, three basic types of bounded travelling waves could occur for a PDE, which are periodic waves, kink waves, and solitary waves. Sometimes, they are also called periodic wave trains, fronts, and pulses, respectively. Recall that heteroclinic orbits are trajectories which have two distinct equilibria as their

Motivated by the reasons above, we try to seek all bounded travelling waves of the RLW-Burgers equation and investigate their dynamical behaviors. By some techniques including analyzing the

#### 2. Preliminaries

It is well known that the travelling wave solution has the form

In the following discussion, without loss of generality, we only need to consider the case

System (2.3) has two equilibria

Obviously,

As a special case, when

By the properties of planar Hamiltonian system, we know there is a unique homoclinic orbit

The homoclinic orbit

#### 3. The Existence and Uniqueness of Bounded Travelling Waves

By the Bendixon Theorem, for system (2.3), the expression

Theorem 3.1. *Suppose that *

* Proof. *In the case

From the vector field defined by (2.3), orbits in first quadrant can only go right when

Assume that

The denominator

On the line segment

Now, we can see that

In the case

Then by another Poincaré transformation

Letting

Further, we need to judge the behaviors of orbits in

Next, we prove the existence of a saddle-focus heteroclinic orbit. In fact, from [16], there exist four invariant manifolds near the saddle

In addition, from Figure 3, one can check that the unstable manifold

In fact, the saddle-focus heteroclinic orbit shown by us corresponds to the oscillatory kink wave. These points on the right (left) hand side of focus

Theorem 3.2. *For the oscillatory kink wave in Theorem 3.1, the maximal oscillation amplitude of it is increasing with respect to the parameter *

*Proof. *Let

Assume that

Construct two functions

#### 4. Explicit Expressions of Monotone and Oscillatory Kink Waves

It is difficult to give all exact expressions of the monotone kink waves under the conditions required in Theorem 3.1. But for some special case, for example,

Next, we will apply the extended tanh-function method [18] to deal with the problem. Firstly, we guess that the monotone kink wave can be expressed as a finite series of tanh function. Noting that the fact that the Riccati equation:

Substituting (4.2) into (2.2) and replacing

One can check that

In contrast to the monotone kink wave, it is more difficult to give the exact expression of the oscillatory kink wave solution. Even when

Next, we only need to determine the coefficients

#### 5. Results

From the transformation made in Section 2 and the proofs in Section 3, we can obtain complete results about bounded travelling waves of (1.1) under different parameter conditions as listed in Table 1. Furthermore, from these results and Theorem 3.2, we can see, for the oscillatory kink wave solution, the maximum oscillation amplitude increases with respect to

So, taking the case

From the variable transformation

#### Acknowledgments

This work is supported by the Natural Science Foundation of China (No. 11171046 and No.11061039), the Key Project of Educational Commission of Sichuan Province (No. 12ZA224), the Scientific Research Foundation of CUIT (No. CSRF201007), the Scientific Research Platform Projects for the Central Universities (No. 11NPT02) and Academic Leader Training Fund of Southwest University for Nationalities.

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