KAM for the nonlinear wave equation on the circle: a normal form theorem

TL;DR

This paper proves a KAM theorem for infinite-dimensional Hamiltonian systems with multiple eigenvalues, demonstrating the persistence of invariant tori under small perturbations in the context of nonlinear wave equations on the circle.

Contribution

It extends KAM theory to infinite-dimensional systems with multiple eigenvalues, providing conditions for the persistence of invariant tori in nonlinear wave equations.

Findings

Invariant tori persist under small perturbations.

Conditions for non-resonance of frequencies are established.

The theorem applies to Hamiltonian normal forms with block-diagonal matrices.

Abstract

In this paper we prove a KAM theorem in infinite dimension which treats the case of multiple eigenvalues (or frequencies) of finite order. More precisely, we consider a Hamiltonian normal form in infinite dimension:\begin{equation} \nonumberh(\rho)=\omega(\rho).r + \frac{1}{2} \langle \zeta,A(\rho)\zeta \rangle,\end{equation}where $ r \in \mathbb{R}^n $, $\zeta=((p\_s,q\_s)\_{s \in \mathcal{L}})$ and $ \mathcal{L}$ is a subset of $\mathbb{Z}$. We assume that the infinite matrix $A(\rho)$ satisfies $A(\rho)= D(\rho)+N(\rho)$, where $D(\rho) =\operatorname{diag} \left\lbrace \lambda\_{i} (\rho) I\_2,\: 1\leq i \leq m\right\rbrace$ and $N$ is a bloc diagonal matrix. We assume that the size of each bloc of $N$ is the multiplicity of the corresponding eigenvalue in $D$.In this context, if we start from a torus, then the solution of the associated Hamiltonian system remains on that torus. Under certain conditions emitted on the frequencies, we can affirm that the trajectory of the solution fills the torus. In this context, the starting torus is an invariant torus. Then, we perturb this integrable Hamiltonian and we want to prove that the starting torus is a persistent torus. We show that, if the perturbation is small and under certain conditions of non-resonance of the frequencies, then the starting torus is a persistent torus.