Abstract and Applied Analysis

Volume 2015 (2015), Article ID 820916, 14 pages

http://dx.doi.org/10.1155/2015/820916

## Bifurcation of Traveling Wave Solutions for (2+1)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

^{1}College of Applied Science, Beijing University of Technology, Beijing 100124, China^{2}School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China

Received 27 June 2014; Accepted 15 July 2014

Academic Editor: Yonghui Xia

Copyright © 2015 Yanping Ran et al. 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

Four (2+1)-dimensional nonlinear evolution equations, generated by the Jaulent-Miodek hierarchy, are investigated by the bifurcation method of planar dynamical systems. The bifurcation regions in different subsets of the parameters space are obtained. According to the different phase portraits in different regions, we obtain kink (antikink) wave solutions, solitary wave solutions, and periodic wave solutions for the third of these models by dynamical system method. Furthermore, the explicit exact expressions of these bounded traveling waves are obtained. All these wave solutions obtained are characterized by distinct physical structures.

#### 1. Introduction

In [1–4], four (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy were developed. These nonlinear models are completely integrable evolution equations. There are many approaches to investigate nonlinear evolution equation, for example, the inverse scattering method, the Bäcklund transformation method, the Darboux transformation method, the Hirota bilinear method [1–3, 5–8], and the dynamical systems method [9–11]. The Hirota bilinear method [3] is used to formally derive the multiple kink solutions and multiple singular kink solutions of these models. By applying the direct symmetry method [4], group invariant solutions and some new exact solutions of the (2+1)-dimensional Jaulent-Miodek equation are obtained. Dynamical systems method is a very effective method to research qualitative behavior for traveling wave solutions of these completely integrable evolution equations. In [11], only considering bifurcation parametric , some exact traveling wave solutions are given by applying the method of dynamical systems for these models. In this paper, all wave solutions are given by the method of dynamical systems under more general parametric conditions. Some computer symbolic systems such as Maple and Mathmatic allow us to perform complicated and tedious calculations.

Four (2+1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy [3] are given bywhere is the inverse of with andWe will study the third model given by

By introducing the potentialto remove the integral term in the system (3), we obtain the following equation

We are interested in the wave solutions of the system (3) in this paper. Motivated by [9], we obtain dynamical properties of (11) and different wave solutions of the system (3) in detail. This paper is organized as follows. In Section 2, we establish the traveling wave equation (3) for the third model of (1). Furthermore, we obtain the first integral of dynamical governing equation of the system (11). Then, we analyze the bifurcation behaviors of the system (11). Phase portraits in the different subsets of parameter space will be presented in Section 3. In Section 4, using the information of the phase portraits in Section 3, we analyze all the possible traveling wave solutions of the system (11). Some explicit parametric representations of traveling wave solutions of (3) and the system (11) are also obtained. The final section includes brief summary, future plans, and potential fields of applications.

#### 2. Traveling Wave Equation for the System (3)

We assume that the traveling wave transform of the system (3) is in the formwhere is propagating wave velocity. Let , , the traveling wave transform of (6) is equivalent to [11]. So, our traveling wave transform is more general. According to physical meaning of traveling wave solutions of the system (3), we always assume that , , and . Now, substituting (6) into (5), we have the traveling wave equation

Integrating (7) with respect to once, we haveSetting , (8) becomesFurthermore, (8) can be rewritten as

Letting , then we have the following planar systemObviously, the above system (11) is a Hamiltonian system with Hamiltonian function

In order to research the system (11), let , ; the system (11) becomesThe Hamiltonian function of (13) is

#### 3. The Bifurcation Analysis of the System (11)

In this section, our aim is to study the traveling wave solutions of the system (11) by applying bifurcation method and qualitative theory of dynamical systems [9, 10]. Through some special phase orbits, we obtain smooth periodic wave solutions, solitary wave solutions, kink and antikink wave solutions, and so on. Fixing , we discuss the phase portrait of the system (11) along with the changes of parameters and so as to study traveling wave solutions of the system (11). Further more, through the traveling wave solutions of the system (11) and the potential relation (4), traveling wave solutions of the system (3) will be obtained.

##### 3.1. Phase Portraits and Qualitative Analysis of the System (11)

In order to investigate the phase portrait of the system (11), we set

Let . Obviously, has at least one zero point . The number of the singular points of the system (11) may be decided by the sign of . Obviously, the system (11) has only one trivial singular point . Thus the other singular points of the system (11) are given as follows. (1) When , the system (11) has only one trivial singular point ; (2) when , the system (11) has two singular points , where ; (3) when , the system has a second-order singular point , where .

We notice that the Jacobian of linearized system of the system (11) at the singular points is given byThus, the characteristic values of linearized system of the system (11) at are . From the qualitative theory of dynamical system, we know that(i)if , is a saddle point;(ii)if , is a center point;(iii)if , is a degenerate saddle point.

Letwhere is Hamiltonian value. When ,has four real roots.

It is well known that the planar Hamiltonian system is determined by its potential energy level curve and its singular point in the form of . So, we are interested in looking for the possible zeros of (15) and determining whether there are heteroclinic orbits, homoclinic orbits, periodic orbits at different singular points.

In order to find the heteroclinic orbits and the homoclinic orbits of the system (11), letFrom (19), we can get the following expressions of its roots:

Substituting (20) into (15), we can get

Theorem 1. *When , , from (22), one has the following.*(i)*When , there are two heteroclinic orbits formed by the saddle points .*(ii)*When , there are no heteroclinic orbits, while there are homoclinic orbits formed by other saddle points except for two saddle points .*

*Proof. *When , we have and . According to the qualitative theory of dynamical system, are saddle points. Furthermore, when , holds. Similarly, if , we have that is the saddle point and holds. Applying Theorems 1 and 2 [12], Theorem 1 is proved.

*In order to give the details of the bifurcation, if , , we can obtain the following six bifurcation boundaries:*

*All these bifurcation boundaries divide the parameter space into seven regions (see Figure 1(a)) in which different phase portraits exist. All the corresponding phase portraits will be shown in Figure 2.*