 18116 Massimiliano Berti, Roberto Feola, Fabio Pusateri
 Birkhoff normal form and long time existence
for periodic gravity water waves.
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Nov 17, 18

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Abstract. We consider the gravity water waves system with a periodic onedimensional 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 ZakharovDyachenko [62] based on the formal Birkhoff
integrability of the water waves Hamiltonian truncated at order four. As a consequence, we also
obtain a longtime 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 nearresonances, and nontrivial resonant fourwaves interactions, the socalled
BenjaminFeir 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 BenajaminFeir resonances by
comparing with the formal computations of [62, 22, 30, 20].
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