Exact Solitary Wave Solution in the ZK-BBM Equation

The traveling wave solution for the ZK-BBM equation is considered, which is governed by a nonlinear ODE system. The bifurcation structure of fixed points and bifurcation phase portraits with respect to the wave speed c are analyzed by using the dynamical system theory. Furthermore, the exact solutions of the homoclinic orbits for the nonlinear ODE system are obtained which corresponds to the solitary wave solution curve of the ZK-BBM equation. 1. Introduction Nonlinear dispersive equations are important models to describe a lot of physical phenomena and engineering problems. Among all the nonlinear phenomena exhibited by the systems, the solitary wave is one of the most interesting motions, which is a special wave related to many physical and mathematical problems such as turbulence and chaos. But it is usually not a simple work to find the solitary wave in a nonlinear dispersive equation. Several methods have been introduced to find a solitary wave in those equations, such as the tanh-sech method, the sine-cosine algorithm, the homogeneous balance method, and the inverse scattering method. See [1–3] for details. Among all the nonlinear dispersive systems, the KdV equation and the dissipative Burgers equation have been paid more attention by many authors and their general wave solution and the solitary wave solution have been well discussed. See [4, 5] for details. Some general forms of the KdV equation have also been introduced. In 1972, the Benjamin-Bona-Mahony equation has been proposed as a model for propagation of long waves, where the nonlinear dispersion is incorporated. See [6] for details. Its assumption is similar to KdV equation and the model is as follows: In 1974, Zakharov and Kuznetsov proposed an equation to govern the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and hot isothermal electrons. See [7] for details. The ZK equation is In 2005, Wazwaz structured an equation by combining the BBM equation with the ZK equations, that is, the ZK-BBM equation: The dynamics of the ZK-BBM equation has been discussed and the existence of the solitary wave has been considered in that paper. But the exact solutions of the solitary wave are still unknown, which would be more important to find out other dynamics of the ZK-BBM equation. In the paper, we will attempt to find the solitary wave in the ZK-BBM equation. The traveling wave for this equation, , is considered in this study, which is governed by a nonlinear ODE system, whose homoclinic orbit is just the solitary wave of the ZK-BBM equation. The bifurcation