Abstract

We consider the water wave equations for a 2D ocean of finite depth under the action of gravity. We present a recent existence and linear stability result for small amplitude standing wave solutions that are periodic in space and quasi-periodic in time. The result holds for values of a normalized depth parameter in a Cantor-like set of asymptotically full measure. 
The main difficulties of the problem are the presence of derivatives in the nonlinearity (the system is quasi-linear), and a small divisors problem where the frequencies of the linear part grow in a sublinear way at infinity (like the square root of integers). To overcome these problems we first reduce the linearized operators (which are obtained at each approximate quasi-periodic solution along a Nash-Moser iteration) to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak second Melnikov non-resonance conditions (losing derivatives both in time and space). Such non-resonance conditions are sufficiently weak to be satisfied for most values of the normalized depth parameter, thanks to arguments from degenerate KAM theory.
Joint work with Massimiliano Berti, Emanuele Haus and Riccardo Montalto.