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.

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