Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves |
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Authors: | Massimiliano Berti Roberto Feola Fabio Pusateri |
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Institution: | 1. SISSA, Via Bonomea 265, Trieste, 34136 Italy;2. Departimento di Matematica e Fisica, Universitá degli Studi ROMA TRE Largo San Leonardo Murialdo, 1-00146, Roma, Italy;3. University of Toronto, 40 St. George St., Room 6218, Toronto, ON, M5S 2E4 Canada |
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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 . This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. |
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