We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko-Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order (Formula presented.). This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.
Berti, M., Feola, R., Pusateri, F. (2022). Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS [10.1002/cpa.22041].
Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves
Berti M.;Feola R.;Pusateri F.
2022-01-01
Abstract
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko-Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order (Formula presented.). This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.File | Dimensione | Formato | |
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