Kam Theory And Its Applications In Differential Equations

We are mainly interested in KAM theory,and apply KAM theory to prove that there exists a positive quasi-periodic solution for Lotka-Volterra system with quasiperiodic coefficients and there exist 2-dimensional invariant tori both for the cubic complex Ginzburg-Landau equation with higher spatial dimension and 1D Ginzburg-Landau equation with nonlinearity |u|^2pu.This paper is divided into four parts.In Chapter 1,we introduce the historical background,some recent results of KAM theory obtained in the literature and our main work in this paper.In Chapter 2,it is proven that the existence of quasi-periodic solutions for Lotka-Volterra system with quasi-periodic coefficients.Lotka-Volterra system plays an important role in the research field of ecosystem.It includes predator-prey system,competitive system,cooperative systems and host-parasite systems.Many work mostly discuss periodic Lotka-Volterra system.In fact,quasi-periodic system describes our world more realistic and more accurate than periodic one.We use KAM iteration to prove the existence of quasi-periodic solutions for the system with quasi-periodic coefficients.In Chapter 3,we consider the cubic complex Ginzburg-Landau equation with higher spatial dimension.Ginzburg-Landau equation is an important equation with very rich physical and mathematical background.It can be described Banerd convection, Tayor-Couette flow,Plane Poiseuille flow,as well as issues such as chemical turbulence, and also applied to non-equilibrium phase transition,theory of superconductivity and superfluidity.It is proven that the existence of 2-dimensional invariant tori for this equation by a modified KAM theorem.In Chapter 4,it is proven that the existence of 2-dimensional invariant tori for 1D Ginzburg-Landau equation with nonlinearity |u|^2pu by a modified KAM theorem.