Nonlinear Wave Traveling Wave Solutions Of The Dynamical Behavior Of Differential Systems Bifurcation Of Limit Cycles

Nonlinear wave equations, as one type of the partial differential equations, are one of the important branches of nonlinear science. Studying on the methods for finding solutions of nonlinear wave equations is an old but vital research subject. So far, many effective methods have been proposed and developed to search for exact solutions of the nonlinear wave equations, unfortunately, there exists no approach to universally apply to solve all kinds of nonlinear wave equations once and for all. As a result, it is still a challenging and significant task to proceed to search for various powerful and effective approaches to solve the nonlinear wave equations.This doctoral dissertation is devoted to study the traveling wave solutions of some important kinds of nonlinear wave equations from the viewpoint of dynamical systems. The smooth and non-smooth traveling wave solutions and their qualitative behaviors are investigated by applying the qualitative theory and bifurcation theory of differential equa-tions. The influence of the system's singular curves on the structure of solutions is also considered. Moreover, the various parametric conditions for the existence of boundary so-lutions are analyzed, the exact representations of solutions are given and their dynamical behaviors are discussed.The major work of this dissertation is as follows.In chapter1, the historical background and the classic methods for solving nonlin-ear wave equations are summarized. The recent advance on non-smooth solitary waves is retrospected. The relationship between nonlinear wave equations and dynamical sys-tems is presented. Some preliminary knowledge of dynamical systems are given and the main results of the three-step method for solving nonlinear equations, which proposed by professor Li Jibin, are briefly described as well. In the finale, the research background and the research developments of bifurcation of limit cycles of differential systems are reviewed. Particularly, the current research progress and effective methods on the study of limit cycles bifurcated from the nilpotent points are emphasized.In chapter2, the qualitative behavior of the traveling wave solutions in the case of degenerate singular points are considered by using of the qualitative theory of differential equations. Secondly, two types of single peak solitary wave solutions are obtained by setting the partial differential equation under the inhomogeneous condition. Asymptotic analysis and numerical simulations are provided for the cusped solutions of the equation.In chapter3, the traveling wave solutions of a family of fully nonlinear equations K(m,n) are studied. In view of the current literature which is mainly concentrated on the discussion of compacton solutions of this type of equations. In this part, cuspon solutions, smooth solitary solutions and smooth periodic solutions are obtained by employing the three-step method. Specifically, the fact that the solutions corresponding to the singular homoclinic (or heteroclinic) orbits of the regular system are not solitary solutions or non-smooth solutions but smooth periodic solutions, is proved.In chapter4, the bifurcation theory of dynamical systems is applied to the general-ized Camassa-Holm equation. As a result, a number of smooth and non-smooth solutions are obtained by studying the phase portraits of its corresponding traveling wave system. It is pointed out that peakon and valleyon can coexist under some special parametric condi-tion. It is as well be emphasized that the existence of singular straight line is the original reason for the appearance of non-smooth traveling wave solutions. Under different para-metric conditions, various explicit parametric representations of boundary traveling wave solutions are presented by using the method of integration.In chapter5, the center conditions and bifurcation of limit cycles of a class of quar-tic system having a nilpotent singular point are studied. The quasi-Lyapunov constant is defined and the linear recursive formulas to compute the quasi-Lyapunov constants are given. By the computer algebra system Mathematica, the first nine quasi-Lyapunov con-stants are deduced. the existence of nine small-amplitude limit cycles generated from the nilpotent singular point is proved.In chapter6, the bifurcation of limit cycles of a class of six degree system having a nilpotent singular point is investigated. The recursive formulas for the computation of the quasi-Lyapunov constants are presented, by which the first thirteen quasi-Lyapunov constants are deduced. Furthermore, the conditions for the nilpotent singular point to be a thirteen-order fine focus are derived. It is for the first time that a system of six degree which can bifurcate thirteen limit cycles from the nilpotent singular point is constructed.