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.

%V 3 %P 85 - 115 %8 2021/04/01 %@ 2523-3688 %G eng %U https://doi.org/10.1007/s42286-020-00036-8 %N 1 %! Water Waves %0 Journal Article %D 2021 %T Traveling Quasi-periodic Water Waves with Constant Vorticity %A Massimiliano Berti %A Luca Franzoi %A Alberto Maspero %XWe 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.

%V 240 %P 99 - 202 %8 2021/04/01 %@ 1432-0673 %G eng %U https://doi.org/10.1007/s00205-021-01607-w %N 1 %! Archive for Rational Mechanics and Analysis %0 Journal Article %D 2019 %T Reducibility for a fast-driven linear Kleinâ€“Gordon equation %A Luca Franzoi %A Alberto Maspero %XWe prove a reducibility result for a linear Kleinâ€“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.

%V 198 %P 1407 - 1439 %8 2019/08/01 %@ 1618-1891 %G eng %U https://doi.org/10.1007/s10231-019-00823-2 %N 4 %! Annali di Matematica Pura ed Applicata (1923 -)