KAM POUR L’ÉQUATION DES ONDES SUR LE CERCLE: UN THÉORÈME DE FORME NORMALE

Bouthelja, Moudhaffar

Document type :

Pré-publication ou Document de travail

Title :

KAM POUR L’ÉQUATION DES ONDES SUR LE CERCLE: UN THÉORÈME DE FORME NORMALE

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 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:\begi ...
Show more >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.Show less >

Language :

Anglais

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Laboratoire Paul Painlevé - UMR 8524

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