01345nas a2200169   4500008004100000020001400041245006600055210006500121260001500186300001300201490000600214520084700220100002401067700001901091700001801110856004701128       2021                        eng d  a2523-368800aQuadratic Life Span of Periodic Gravity-capillary Water Waves0 aQuadratic Life Span of Periodic Gravitycapillary Water Waves  c2021/04/01  a85 - 1150 v33 a<p>We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible presence of three-wave resonances for general values of gravity, surface tension, and depth, such normal form may be not trivial and exhibit a chaotic dynamics (Wilton ripples). Nevertheless, we prove that for all the values of gravity, surface tension, and depth, initial data that are of size $$ \varepsilon $$in a sufficiently smooth Sobolev space leads to a solution that remains in an $$ \varepsilon $$-ball of the same Sobolev space up times of order $$ \varepsilon ^{-2}$$. We exploit that the three-wave resonances are finitely many, and the Hamiltonian nature of the Birkhoff normal form.</p>1 aBerti, Massimiliano1 aFeola, Roberto1 aFranzoi, Luca  uhttps://doi.org/10.1007/s42286-020-00036-8