Full-dimensional KAM torus with frequency-preserving in infinite-dimensional Hamiltonian systems

TL;DR

This paper proves the existence of full-dimensional invariant tori with prescribed frequencies in infinite-dimensional Hamiltonian systems, including nonlinear Schrödinger equations, using a novel frequency-preserving KAM theorem.

Contribution

It introduces the first Kolmogorov type result for infinite-dimensional systems with frequency preservation, applicable without spectral asymptotics.

Findings

Proves persistence of full-dimensional KAM tori with specified frequencies.

Establishes the first Kolmogorov type theorem in infinite dimensions.

Confirms Bourgain's conjecture on invariant tori for 1D nonlinear Schrödinger equations.

Abstract

In this paper, we present two infinite-dimensional KAM theorems with frequency-preserving for a nonresonant frequency of Diophantine type or even weaker. To be more precise, under a nondegenerate condition for an infinite-dimensional Hamiltonian system, we prove the persistence of a full-dimensional KAM torus with the specified frequency independent of any spectral asymptotics, by advantage of the generating function method. This appears to be the first Kolmogorov type result in the infinite-dimensional context. As a direct application, we provide a positive answer to Bourgain's conjecture: full-dimensional invariant tori for 1D nonlinear Schr\"{o}dinger equations do exist.