KAM for gravity water waves in finite depth

Abstract

We present the recent result in [3] concerning the existence of Cantor families of small amplitude, linearly stable, time quasi-periodic standing water wave solutions – i.e. periodic and even in the space variable $x$ – of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure.