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

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: $$\begin{aligned} h(\rho )=\omega (\rho )\cdot r + \frac{1}{2} \langle \zeta ,A(\rho )\zeta \rangle , \end{aligned}$$ where $$ r \in \mathbb {R}^n $$ , $$\zeta =((p_s,q_s)_{s \in \mathcal {L}})$$ and $$ \mathcal {L}$$ is a subset of $$\mathbb {Z}$$ and $$\rho $$ is a parameter that belongs to a compact. We assume that the infinite matrix $$A(\rho )$$ is of the form $$A(\rho )= D(\rho )+N(\rho )$$ , where $$D(\rho )$$ is a diagonal matrix with finite eigenvalue multiplicity 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 prove that, if the perturbation is small and under certain conditions of non-resonance of the frequencies, then the starting torus is a persistent torus. We use this result to prove the existence of quasi-periodic solutions for the nonlinear wave equation with convolutive potential on the circle. In Bouthelja (KAM for the nonlinear wave equation on the circle: small amplitude solutions (preprint), arXiv:1712.01597 , 2017) we use this result to prove the existence of such solutions for the nonlinear wave equation on the circle without an external parameter.