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

Bouthelja, Moudhaffar

Document type :

Pré-publication ou Document de travail

Title :

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

Author(s) :

Bouthelja, Moudhaffar [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

In this paper we consider the nonlinear wave equation on the circle:\begin{equation} \nonumberu_{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 ...
Show more >In this paper we consider the nonlinear wave equation on the circle:\begin{equation} \nonumberu_{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.Show less >

Language :

Anglais

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

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