A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications

TL;DR

This paper extends KAM theory to infinite-dimensional Hamiltonian systems with finite zero normal frequencies, establishing conditions for the existence of KAM tori and applying results to the nonlinear Schrödinger equation.

Contribution

It introduces a new KAM theorem for systems with finite zero frequencies and demonstrates its application to nonlinear Schrödinger equations.

Findings

Existence of KAM tori depends on a specific quantity being zero or not.

Most frequencies lead to either the presence or absence of KAM tori.

The nonlinear Schrödinger equation with zero frequency has many quasi-periodic solutions.

Abstract

In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori or if the quantity is not zero, there is no KAM tori in some domain. As application, we show that the nonlinear Schr\"{o}dinger equation with a zero frequency possesses many quasi-periodic solutions.