@article {2021,
	title = {Quadratic Life Span of Periodic Gravity-capillary Water Waves},
	volume = {3},
	year = {2021},
	month = {2021/04/01},
	pages = {85 - 115},
	abstract = {<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>},
	isbn = {2523-3688},
	url = {https://doi.org/10.1007/s42286-020-00036-8},
	author = {Massimiliano Berti and Roberto Feola and Luca Franzoi}
}
