 1934 M. Berti, R. Feola, L. Franzoi
 Quadratic life span
of periodic gravitycapillary water waves
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May 15, 19

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Abstract. We consider the gravitycapillary water waves equations for a bidimensional fluid
with a periodic onedimensional free surface. We prove a rigorous reduction of this system
to Birkhoff normal form up to cubic degree. Due to the possible presence of 3waves
resonances for general values of gravity, surface tension and depth, such normal form may
be not trivial and exhibit a chaotic dynamics (Wiltonripples). Nevertheless we prove that
for all the values of gravity, surface tension and depth, initial data that are of size \epsilon in a
sufficiently smooth Sobolev space lead to a solution that remains in an \epsilonball of the same
Sobolev space up times of order \epsilon^{2}. We exploit that the 3waves resonances are finitely
many, and the Hamiltonian nature of the Birkhoff normal form.
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