Massimiliano Berti, Roberto Feola, Fabio Pusateri
Birkhoff normal form and long time existence
for periodic gravity water waves.
(1472K, PDF)
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].