Time quasi-periodic gravity water waves in finite depth
| dc.contributor.area | Mathematics | en_US |
| dc.contributor.author | Baldi, Pietro | |
| dc.contributor.author | Berti, Massimiliano | |
| dc.contributor.author | Haus, Emanuele | |
| dc.contributor.author | Montalto, Riccardo | |
| dc.date.accessioned | 2017-09-27T15:46:10Z | |
| dc.date.available | 2017-09-27T15:46:10Z | |
| dc.date.issued | 2017 | |
| dc.description.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 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. | en_US |
| dc.identifier.uri | https://openscience.sissa.it/handle/1963/35296 | |
| dc.language.iso | en | en_US |
| dc.relation.firstpage | 1 | en_US |
| dc.relation.ispartofseries | arXiv;1708.01517 | |
| dc.relation.lastpage | 127 | en_US |
| dc.subject | Water waves | en_US |
| dc.subject | KAM for PDEs | en_US |
| dc.subject | uasi-periodic solutions | en_US |
| dc.subject | standing waves | en_US |
| dc.title | Time quasi-periodic gravity water waves in finite depth | en_US |
| dc.type | Preprint | en_US |