18-116 Massimiliano Berti, Roberto Feola, Fabio Pusateri

Birkhoff normal form and long time existence for periodic gravity water waves. (1472K, PDF) Nov 17, 18

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Abstract. 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 Bikhoff normal form up to degree four. This proves a conjecture of Zakharov-Dyachenko [62] based on the formal Birkhoff integrability of the water 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 ε^{−3}. Main difficulties in the proof are the quasilinear nature of the equations, the presence of small divisors arising from near-resonances, and non-trivial resonant four-waves interactions, the so-called Benjamin-Feir resonances. The main ingredients that we use are: (1) various reductions to constant coefficient operators through flow conjugation techniques; (2) the verification of key algebraic properties of the gravity water waves system which imply the integrability of the equations at non- negative orders; (3) smoothing procedures and Poincare'-Birkhoff normal form transformations; e (4) a normal form identification argument that allows us to handle Benajamin-Feir resonances by comparing with the formal computations of [62, 22, 30, 20].

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