Quasi-periodic Solutions For The Higher Dimensional Cubic Schr(o|")dinger Equations

In this article, We prove an infinite dimensional KAM theorem. As an application, we use the theorem to study the higher dimensional nonlinear Schrodinger equation iut-Δu+|u|2u= 0, t∈R,x∈Td with periodic boundary conditions. We obtain for the equation a Whitney smooth family of small-amplitude quasi-periodic solutions with two frequencies.In the first chapter, we introduce the background of the research, after that we in-troduce the admissible set and give the conclusion of this paper; In the second chap-ter, we give the infinite dimensional KAM theorem; In the third chapter, we verify the Schrodinger equation satisfies the hypothesis of the infinite dimensional KAM theorem, and prove the conclusion of this paper by using the infinite dimensional KAM theorem; In the fourth chapter, through the KAM iteration, the establishment of the infinite dimen-sional KAM theorem is proved.