A Kam Theorem for Space-Multidimensional Hamiltonian PDE

TL;DR

This paper develops an abstract KAM theorem tailored for space-multidimensional Hamiltonian PDEs with smoothing nonlinearities, allowing for hyperbolic components and applications to singular perturbation problems.

Contribution

It introduces a novel KAM theorem accommodating hyperbolic parts in the Hamiltonian and applicable to singular perturbation problems in PDEs.

Findings

The theorem applies to space-multidimensional Hamiltonian PDEs with smoothing nonlinearities.

Invariant tori constructed may be unstable due to hyperbolic components.

The theorem is demonstrated through three applications.

Abstract

We present an abstract KAM theorem, adapted to space-multidimensional hamiltonian PDEs with smoothing non-linearities. The main novelties of this theorem are that: $\bullet$ the integrable part of the hamiltonian may contain a hyperbolic part and as a consequence the constructed invariant tori may be unstable. $\bullet$ It applies to singular perturbation problem. In this paper we state the KAM-theorem and comment on it, give the main ingredients of the proof, and present three applications of the theorem .