Seminari del Dipartimento

Analisi

We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof the conjecture of Zakharov and Dyachenko. More precisely, we provide a reduction of the equations to Birkhoff normal form and prove the integrability of the water waves Hamiltonian up to order four. As a consequence, we also obtain a long-time stability result for periodic solutions: perturbations of a flat interface that are of size $e$  in a standard Sobolev  space lead to solutions that remain regular and small up to times of the order $e^{-3}$.
To our knowledge, this is the first such long-time existence result for quasilinear systems in the periodic setting.Indeed, despite the absence of external parameters, our result goes past the natural $e^{-2}$ scale which one expects for non-quadratically resonant Hamiltonian systems, such as gravity water waves.The main difficulties in the proof are the quasilinear nature of the equations, the presence of small divisors arising from near (trivial) resonances,and of many non-trivial resonant four-way interactions, the so-called Benjamin-Feir resonances.
org: PROCESI Michela