 1783 Pietro Baaldi, Massimiliano Berti, Emanuele Haus, Riccardo Montalto
 Time quasiperiodic gravity water waves
in finite depth
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Aug 8, 17

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Abstract. We prove the existence and the linear stability of Cantor families of small amplitude time
quasiperiodic standing water wave solutions namely periodic and even in the space variable x of a
bidimensional 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 quasilinear 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 quasiperiodic solution along the NashMoser iteration
to constant coefficients up to smoothing operators, using pseudodifferential changes of variables that are
quasiperiodic 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.
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