TY - RPRT
T1 - Time quasi-periodic gravity water waves in finite depth
Y1 - 2017
A1 - Pietro Baldi
A1 - Massimiliano Berti
A1 - Emanuele Haus
A1 - Riccardo Montalto
AB - 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.
UR - http://preprints.sissa.it/handle/1963/35296
U1 - 35602
U2 - Mathematics
ER -