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.

1 aBerti, Massimiliano1 aFeola, Roberto1 aFranzoi, Luca uhttps://doi.org/10.1007/s42286-020-00036-801072nas a2200169 4500008004100000020001400041245006500055210006400120260001500184300001300199490000800212520057200220100002400792700001800816700002100834856004700855 2021 eng d a1432-067300aTraveling Quasi-periodic Water Waves with Constant Vorticity0 aTraveling Quasiperiodic Water Waves with Constant Vorticity c2021/04/01 a99 - 2020 v2403 aWe prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

1 aBerti, Massimiliano1 aFranzoi, Luca1 aMaspero, Alberto uhttps://doi.org/10.1007/s00205-021-01607-w01286nas a2200157 4500008004100000020001400041245006600055210006500121260001500186300001600201490000800217520081700225100001801042700002101060856004701081 2019 eng d a1618-189100aReducibility for a fast-driven linear Klein–Gordon equation0 aReducibility for a fastdriven linear Klein–Gordon equation c2019/08/01 a1407 - 14390 v1983 aWe prove a reducibility result for a linear Klein–Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving; however, we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time-independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which moves the original equation in a perturbative setting. Then, we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions.

1 aFranzoi, Luca1 aMaspero, Alberto uhttps://doi.org/10.1007/s10231-019-00823-2