Invariant tori for commuting Hamiltonian PDEs

TL;DR

This paper extends Nekhoroshev's theorem to certain PDEs, demonstrating the persistence of invariant tori with parameters in a Cantor set, including resonant cases, with applications to nonlinear wave and beam equations.

Contribution

It generalizes the persistence of invariant tori to Hamiltonian PDEs with multiple integrals of motion, using Lyapunov-Schmidt decomposition and implicit function theorem.

Findings

Persistence of r-parameter family of invariant tori in PDEs

Construction of 2D tori for nonlinear wave equations

Construction of 3D tori for coupled beam equations

Abstract

We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an $r$-parameter family of $r$-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2 dimensional tori, while in the second case we construct 3 dimensional tori.