Traveling Front Solutions Of Several Singularly Perturbed Reaction-diffusion Equations

By combining geometric singular perturbation theory and Melnikov function,this thesis studies several singularly perturbed reaction-diffusion systems including a real supercritical Ginzburg-Landau equation coupled by a slow diffusion mode,the FitzHugh-Nagumo equation with jump-heterogeneity and a sine-Gordon equation with two slowly varying parameters,where the slow and fast dynamics axe separated and then the fast and slow orbits are connected transversally.The thesis is divided into five chapters:Chapter 1 is the introduction.In this chapter,we introduce singular perturba-tion geometry theory and some recent researches within this subject as well as the content of this thesis.In chapter 2,we study the existence and bifurcation of traveling front solu-tions in a singularly perturbed reaction-diffusion system coupled by supercritical Ginzburg-Landau equation and a slow diffusion mode.It is shown that after the coupling of the slow diffusion mode,the traveling fronts of supercritical Ginzburg-Landau equation persists and undergoes heteroclinic saddle-node bifurcation under appropriate parameter value.In chapter 3,we study the existence of pinned 1-front in a FitzHugh-Nagumo equation with jump-heterogeneity.Based on geometric singular perturbation theo-ry,a fast orbit of the layer system and two slow orbits of the degenerate system are connected transversally,to define singular heteroclinic orbit with certain transver-sality.After perturbation,we obtain the existence of pinned 1-front and determine the position of pinning.In chapter 4,we study the existence of 1-pulse heteroclinic orbit which is het-eroclinic to two saddles on the slow manifolds in a sine-Gordon equation with two slowly varying parameters.Chapter 5 summarizes the researches in the thesis and gives some future works.