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.

%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 -)