Study On The Bifurcations And The Dynamics Of Traveling Wave Solutions For Several Classes Of Nonlinear Equations

With the development of nonlinear science, nonlinear equations has become one of thecore problem in the study of nonlinear science. This paper mainly uses the bifurcation theory of dynamical systems to investigate the bifurcation and dynamical behavior of travelling wave solutions for several classes of nonlinear equations. We analysis the difference between kink wave solution and kink-compacton solution from the angle of dynamical systems for the first time.In Chapter 1, the research background, the research status of the nonlinear wave equations and the main work are summarized.In Chapter 2, we mainly introduce some preliminary knowledges and the bifurcation theory of the nonliear traveling wave equation.In Chapter 3, we investigate traveling wave solutions of a nonlinear wave equation with degenerate dispersion. The phase portraits of corresponding traveling wave system are given under different parametric conditions. Some periodic wave and smooth solitary wave solutions of the equation are obtained. Moreover, we find some new hyperbolic function compactons instead of well-known trigonometric function compactons by analyzing nilpotent points.In Chapter 4, by using the method of dynamical systems, we investigate solitary wave,kink wave and kink-compacton solutions of a modified4? model. Under different parametric conditions, the phase portraits of the associated traveling wave system are given,and the parametric representations of three types of traveling wave solutions are obtained.Moreover, we analysis the difference between kink wave solution and kink-compacton solution from the angle of dynamical systems for the first time.In Chapter 5, we summarize the work of the paper and put forward the research work in the future.