Researches On Bifurcation And Nonlinear Waves In Several Singularly Perturbed Systems

By combining geometric singular perturbation theory with blow-up technique,Melnikov method and phase plane analysis method,etc.,this paper is devoted to studying the bifurcation and nonlinear waves in several singularly perturbed system-s,including the bifurcation of "singular" transcritical point in a planar singularly perturbed system,the existence of new solitary wave solutions to a perturbed gener-alized Beniamin-Bona-Mahony(BBM)equation,and the existence of travelling wave solutions in the perturbed and unperturbed Degasperis-Procesi(DP)equation.The article is divided into five chapters:Chapter 1 is the introduction of the thesis,in which,we introduce the research background and the structure of this article as well as the geometric singular per-turbation theory.In Chapter 2,based on the geometric singular perturbation theory and the blow up technique,we study the bifurcation of "singular" transcritical point in a planar singularly perturbed system.By analyzing the global dynamics of the layer system(?=0),the orbits on the charts K1(x=1)and K2(?=1),and matching and connecting the orbits between these charts,this paper reveals the bifurcation phe-nomenon of the plane singularly perturbed the corresponding bifurcation parameter curves.In Chapter 3,by combining geometric singular perturbation theory with phase plane analysis and explicit Melnikov method,we study the existence of new solitary wave solutions in a perturbed generalized BBM equation.Firstly,by the traveling wave transform and Fenichel's first invariant manifold theorem,the perturbed gen-eralized BBM equation is transformed into a planar Hamiltonian-perturbed system.Then,the geometric singular perturbation theory and the explicit Melnikov method are introduced to obtain the new solitary wave solutions,which are homoclinic to the non-trivial steady states.Different from the previous work,we assume that the integral constant is non-zero,so the solitary wave solutions are new.In Chapter 4,based on the singular traveling wave method and the geometric singular perturbation theory,we study the existence and classification of various traveling wave solutions in a the D-P equation as well as the existence of solitary wave solutions to the perturbed D-P Equation.Firstly,by using the traveling wave transform,the D-P equation is transformed into a planar systems with a singular straight line whose associated regular system is a Hamiltonian system.By ana-lyzing the global phase portrait of the Hamiltonian system,the global dynamics of the singular planar system are obtained,i.e.,the existence and classification of various traveling wave solutions in the D-P equation are derived.Next,we use the explicit Melnikov method to consider the existence of solitary wave solutions in the perturbed D-P equation i.e.,the persistence of the solitary wave solutions under sin-gular perturbation.Since the Melnikov integral is explicitly,so the speed of solitary wave solution to leading order is determined.In Chapter 5,we summary this thesis and give some problems to future work.