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

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 [

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 [

In the paper, we will attempt to find the solitary wave in the ZK-BBM equation. The traveling wave for this equation,

This paper is arranged as follows. In Section

Assume that the nonlinear ZK-BBM equation

When the constant

For

When

When

When

The bifurcation phase portraits of system (

The bifurcation phase portraits of system (

By the above analysis, system (

Using the ODE

For the wave speed

By the bifurcation analysis, system (

Similarly, we obtain the integral form of the homoclinic orbit as follows:

In order to obtain solution of (

Substituting rational linear transformation

The conversion conditions of transforming (

It is easy to examine that the coefficients of

For the wave speed

The bifurcation of phase portraits of system (

For

When

When

When

The bifurcation phase portraits of system (

The bifurcation phase portraits of system (

By the bifurcation analysis, system (

It is easy to examine that the coefficients of

For the wave speed

In this paper, the traveling wave for ZK-BBM equation is considered. The bifurcation phase portraits of nonlinear system governing the traveling wave were studied with respect to the wave speed

It is valuable to point out that the method used in this paper can be widely applied to other nonlinear equations with the similar types. We provide the idea to handle complex integral in order to get the exact solution of homoclinic orbits. Therefore, it lays the foundation for studying the chaotic conditions of ZK-BBM equation in the future.

The authors declare that there is no conflict of interests regarding the publication of this paper.