We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth, and prove a rigorous reduction of these equations to Birkhoff normal form up to degree four. This prove a conjecture of Zakharov-Dyachenko based on the formal Birkhoff integrability of the waver waves Hamiltonian truncated at order four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size ε in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order $\epsilon^{−3}$.
Birkhoff normal form and long time existence for periodic gravity water waves
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Nom de l'orateur
Roberto Feola
Etablissement de l'orateur
LMJL
Date et heure de l'exposé
Lieu de l'exposé
Salle des seminaires