Pietro Baaldi, Massimiliano Berti, Emanuele Haus, Riccardo Montalto
Time quasi-periodic gravity water waves 
in finite depth
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ABSTRACT.  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 nonresonance 
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.