%0 Report
%D 2017
%T Time quasi-periodic gravity water waves in finite depth
%A P Baldi
%A Massimiliano Berti
%A Emanuele Haus
%A Riccardo Montalto
%X We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely 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. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the 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 Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.
%G en
%U http://preprints.sissa.it/handle/1963/35296
%1 35602
%2 Mathematics
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-09-27T15:46:10Z
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