TY - JOUR T1 - Quadratic Life Span of Periodic Gravity-capillary Water Waves Y1 - 2021 A1 - Massimiliano Berti A1 - Roberto Feola A1 - Luca Franzoi AB -

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.

VL - 3 SN - 2523-3688 UR - https://doi.org/10.1007/s42286-020-00036-8 IS - 1 JO - Water Waves ER - TY - JOUR T1 - Traveling Quasi-periodic Water Waves with Constant Vorticity Y1 - 2021 A1 - Massimiliano Berti A1 - Luca Franzoi A1 - Alberto Maspero AB -

We 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.

VL - 240 SN - 1432-0673 UR - https://doi.org/10.1007/s00205-021-01607-w IS - 1 JO - Archive for Rational Mechanics and Analysis ER - TY - JOUR T1 - Reducibility for a fast-driven linear Klein–Gordon equation Y1 - 2019 A1 - Luca Franzoi A1 - Alberto Maspero AB -

We 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.

VL - 198 SN - 1618-1891 UR - https://doi.org/10.1007/s10231-019-00823-2 IS - 4 JO - Annali di Matematica Pura ed Applicata (1923 -) ER -