Traveling Wave Solutions For A Few Nonlinear Equations

A fractional or integer order nonlinear partial differential equation is one of the math-ematical models which are used to describe the basic principles of nature phenomena. For the models, the research of finding the representations and investigating the dynamics of their traveling wave solutions is useful to obtain the law of behavior for the system’s motions.In this thesis, taking some nonlinear fractional or integer order equations which come from physical models as objects of research, parameter representations and qualitative dy-namical behavior of some traveling wave solutions of these equations are obtained, which reveal the rich dynamics contained by these nonlinear models. To find the traveling wave solutions expressed by power functions for some fractional partial differential equations, a homogenous principle is obtained and used to get the solutions for some given equa-tions. For some integer order partial differential (Konopelchenko-Dubrovsky) equations with regular traveling systems, the bifurcation theory method of planar dynamical sys-tems is employed to find the bounded smooth traveling wave solutions. For the given rod integer order equations with singular traveling systems, the "three-step method " based on the bifurcation theory method of planar dynamical systems is used to investigate kink wave solutions, periodic wave solutions and unbounded solutions under some parameter conditions.There are five chapters in this thesis.In Chapter1, the research background and main methods of investigating traveling wave solutions for nonlinear equations are summarized. There are two sections in this chapter. In Section1, for fractional order nonlinear partial differential equations, the his-torical background, basic theory, the examples for applications and present situation of solving these equations are summarized. In Section2, for integer order nonlinear par-tial differential equations with regular or singular traveling wave systems, the dynamical system method for finding traveling wave solutions is introduced.In Chapter2, the space-time scaling invariant traveling wave solutions of some frac- tional equations are investigated. Firstly, for finding the solutions, a basic principle, i.e. homogenous principle is obtained. Secondly, the principle is used, for generalized frac-tional Benjamin-Ono equations and generalized fractional Zakharov-Kuznetsov equations in plasma, to obtain the solutions which are expressed by power functions under some pa-rameter conditions.In Chapter3, for the (2+1) dimensional Konopelchenko-Dubrovsky equation, the bifurcation theory method of planar dynamical systems is employed to find the bounded smooth traveling wave solutions:solitary wave solutions, kink (anti-kink) wave solutions, periodic wave solutions. The parameter conditions which make the solutions exist and the exact explicit parametric representations of12solutions are obtained.In Chapter4, for the nonlinear wave equation of longitudinal oscillation in a nonlin-ear elastic rod with lateral inertia, the "three-step method " from the view of the bifur-cation theory method of planar dynamical systems is used to investigate kink wave solu-tions, periodic wave solutions and unbounded solutions under some parameter conditions. Parametric representations of52solutions are obtained. From the qualitative behavior of the solutions, it is that the dynamical behavior of longitudinal oscillation, for the given rod, becomes more and more complex along with the enhancement of nonlinearity for the material.In Chapter5, the summary of this thesis and the prospect of future research are given.