Nonlinear Differential Equation Of The Traveling Wave Solution Branch

In this paper, the bifurcation behavior for a class of nonlinear evolution equations is studied by using the method of the bifurcation theory of dynamical systems. The bifurcation sets and phase portraits of the travelling wave equation in the different regions of the parameter space are given. The different parameter conditions for the existence of solitary wave solutions and kink wave solutions are rigorously determined. Explicit formulas of solitary wave solutions and kink wave solutions are given partly. At the same time, by employing the theory of bifurcations of dynamical systems to a modified Camassa-Holm equation, the existence of solitary wave solutions, uncountably infinite many smooth and non-smooth periodic traveling wave solutions, is obtained in the given areas of parametric space.