Dynamic Analysis For The Nonlinear ZK-BBM Equation

The nonlinear equation often is described to the problems of the engineering science as a mathematical model. It is important to research solution of equation and its dynamic. Nonlinear ZK-BBM equation is a solitary wave equation coupled by the ZK equation and the BBM equation, which has been widely studied.In this paper, the traveling wave for ZK-BBM equation is considered. The bifurcation phase portraits of nonlinear system governing the traveling wave are studied with respect to the wave speed c. Homoclinic orbits are obtained explicitly for some parameter conditions. Thus the exact solitary wave solutions of ZK-BBM equation are obtained. At the same time, we give the wave crest value in the different case of wave speed c.The disturbed ZK-BBM equation is considered, which is governed by a nonlinear ODE system with damping perturbation and external excitation disturbance. The chaotic threshold curve of system with perturbation is analyzed by using the Melnikov theory, which is carried out to compute the distance of stable manifold and unstable manifold of homoclinic orbit in the disturbed system. The chaos conditions of system in the sense of Smale horseshoe in Poincare mapping is gained. Furthermore, through using numerical simulation, the period doubling bifurcation, phase diagrams, time-series diagrams, Poincare mapping diagrams and Lyapunov exponents diagrams with respect to perturbation parameter is obtained, and the corresponding simulation is carried out for specific parameters.