Birkhoff Normal Form and Hamiltonian PDEs

TL;DR

This paper discusses the use of Birkhoff normal form techniques to analyze the long-term stability of solutions in Hamiltonian PDEs, extending finite-dimensional results to infinite dimensions.

Contribution

It introduces an abstract Birkhoff normal form theorem for Hamiltonian PDEs, enabling new insights into their stability and long-time behavior.

Findings

Extended Birkhoff normal form to infinite-dimensional Hamiltonian systems

Provided stability results for Hamiltonian PDEs

Illustrated techniques with examples of PDEs

Abstract

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centered on the Birkhoff normal form theorem that we first proved in finite dimension. Then, after giving some exemples of Hamiltonian PDEs, we present an abstract Birkhoff normal form theorem in infinite dimension and discuss the dynamical consequences for Hamiltonian PDEs.