KAM FOR THE NONLINEAR WAVE EQUATION ON THE CIRCLE: SMALL AMPLITUDE SOLUTION

KAM POUR L' ÉQUATION DES ONDES NON LINÉAIRE SUR LE CERCLE: SOLUTION DE FAIBLE AMPLITIUDE

Moudhaffar Bouthelja (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper we consider the nonlinear wave equation on the circle: \begin{equation} \nonumber u_{tt} - u_{xx} + m u = g(x,u), \quad t \in \mathbb{R},\: x \in \mathbb{S}^1, \end{equation} where $m \in [1,2]$ is a mass and $g(x,u)=4u^3+ O(u^4)$. This equation will be treated as a perturbation of the integrable Hamiltonian: \begin{equation} \tag{$\ast$} \label{first equation} u_t= v, \quad v_t = - u_{xx} + m u. \end{equation} Near the origin and for generic $m$, we prove the existence of small amplitude quasi-periodic solutions close to the solution of the linear equation \eqref{first equation}. For the proof we use an abstract KAM theorem in infinite dimension and a Birkhoff normal form result.

Domaines: Equations aux dérivées partielles [math.AP]

Moudhaffar Bouthelja. KAM FOR THE NONLINEAR WAVE EQUATION ON THE CIRCLE: SMALL AMPLITUDE SOLUTION. 2017. ⟨hal-01654069⟩