KAM for Hamiltonian partial differential equations with weaker Spectral Asymptotics

TL;DR

This paper develops an abstract KAM theorem for infinite-dimensional Hamiltonian PDEs with weaker spectral asymptotics, enabling the proof of many invariant tori and quasi-periodic solutions in high-dimensional equations.

Contribution

It introduces a novel KAM theorem applicable to Hamiltonian PDEs with weaker spectral asymptotics, broadening the scope of equations where invariant tori can be established.

Findings

Existence of many invariant tori in high-dimensional Hamiltonian PDEs.

Construction of quasi-periodic solutions for Schrödinger and Klein-Gordon equations.

Applicability to a wide class of PDEs with weaker spectral asymptotics.

Abstract

In this paper, we establish an abstract infinite dimensional KAM theorem dealing with normal frequencies in weaker spectral asymptotics \Omega_{i}(\xi)=i^d+o(i^{d})+o(i^{\delta}), where $d>0, \delta<0$, which can be applied to a large class of Hamiltonian partial differential equations in high dimensions. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for Schr\"odinger equations, the Klein-Gordon equations with exponential nonlinearity and other equations of any spatial dimension.