Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves

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 eq...

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Published inCommunications on pure and applied mathematics Vol. 76; no. 7; pp. 1416 - 1494
Main Authors Berti, Massimiliano, Feola, Roberto, Pusateri, Fabio
Format Journal Article
LanguageEnglish
Published Melbourne John Wiley & Sons Australia, Ltd 01.07.2023
John Wiley and Sons, Limited
Subjects
Applications of mathematics
Canonical forms
Integral calculus
Integral equations
Interface stability
Mathematical analysis
Periodic variations
Perturbation
Water waves
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Summary: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 ε−3. 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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ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.22041