The Existence Of Traveling Wave Solutions And Solitary Wave Solutions For Some Higher Order KdV Equations And Reaction-diffusion Equations

Dispersion-dissipation equation,high order KdV equation and reaction diffusion equation are several nonlinear differential equations with important significances.In this paper,by using the method of dynamics system,especially geometric singular perturbation theory,combined with Schauder's fixed point theory,the upper and lower solution method and the implicit function theorem and Fredholm theory,we study the traveling wave solution and solitary wave solution of the dispersion-dissipation equation?fifth-order KdV equation and reaction-diffusion predator-prey equation with Allee effect.This paper includes four chapters as follows:Chapter 1 introduces the present situation,backgrounds,significances of our study.Moreover,some famous and basic models of dynamics system are briefly introduced.Chapter 2 introduces some basic concepts and preliminary knowledge.Chapter 3 considers a generalized nonlinear forth-order dispersive-dissipative equation with a nonlocal strong generic delay kernel,which describes wave propagation in generalized nonlinear dispersive,dissipation and quadratic diffusion media.By using geometric singular perturbation theory,Fredholm alternative theory and the linear chain trick,we get a locally invariant manifold and construct the desire heteroclinic orbit.Furthermore we construct a traveling wave solution for the nonlinear equation.Chapter 4 considers the solitary wave solutions in a nonlinear fifth-order KdV equation,which has wide applications in quantum mechanics,nonlinear optics and describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice.Due to geometric singular perturbation theory,we establish the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel.Chapter 5 discusses the traveling wave solutions of a reaction-diffusion predatorprey equation with Allee effect.By applying the Schauder's fixed point theorem as well as upper and lower solution method,we first investigate the existence of traveling wave solutions without delay.Next,we discuss the reaction-diffusion system with a special nonlocal delay convolution kernel by using the method of dynamical system,especial the geometric singular perturbation theory and invariant manifold theory.According to the relationship between traveling wave and homoclinic orbit,the reaction-diffusion equation with Allee effect can be transformed into the ordinary differential system owing to the variable substitution.We construct a locally invariant manifold and use this manifold to obtain the desire heteroclinic orbit due to linear chains technology and Fredholm theory to establish the existence result of traveling wave solution for the system.