The Time-periodic Solutions For A Nonlinear Evolution Equations

In physics, the time-periodic solutions for a nonlinear evolution equations plays a very important role in the inspection of nonlinear phenomena, more and more scholars are more concerned about the solution of nonlinear evolution equations of the time-periodic solutions. In this paper, we mainly consider the existence of the time-periodic solutions for a nonlinear wave equation and the time-periodic solutions for the nonlinear strain wave equation coupled with nonlinear schrodinger equation by Galerkin method and Leray-Schauder fixed point theorem. This paper is composed of three chapters.In chapter1, we introduce the background of the problem and the main work of the paper and some preparations are given here.In chapter2, we consider the time-periodic solution for the Kuramoto-Sivashin-sky Equation, the existence and uniqueness of a time-periodic solution is proved by the Galerkin method and Leray-Schauder fixed point theorem.In chapter3, we consider the time-periodic solution for the nonlinear strain wave equation coupled with nonlinear schrodinger equation, solution of the problem fail to the compact, we can not used directly the imbed theorem, and we can not obtain existence of the time-periodic solution by using directly Leray-Schauder fixed point theorem. However, firstly, we can construct approximate solutions by the Galerkin method; secondly, we prove the existence of the time-periodic solution by using Leray-Schauder fixed point theorem; finally, we prove that limit of approximate solution is solution of former equation by using a priori estimates and the compact.