An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrodinger equation systems

TL;DR

This paper develops an abstract infinite dimensional KAM theorem and applies it to prove the existence of small amplitude quasi-periodic solutions in high-dimensional nonlinear Schrödinger systems with periodic boundary conditions.

Contribution

It introduces a novel abstract KAM theorem for infinite-dimensional systems and demonstrates its application to nonlinear Schrödinger equations with real Fourier multipliers.

Findings

Existence of Whitney smooth small amplitude quasi-periodic solutions.

Construction of a block-diagonal normal form for the system.

Proof of invariant tori in high-dimensional Schrödinger systems.

Abstract

In this paper we consider nonlinear Schrodinger systems with periodic boundary condition in high dimension. We establish an abstract infinite dimensional KAM theorem and apply it to the nonlinear Schrodinger equation systems with real Fourier Multiplier. By establishing a block-diagonal normal form, We prove the existence of a class of Whitney smooth small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.