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 ${\epsilon }^{-3}$. This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

中文翻译：

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周期性重力水波的 Birkhoff 范式和长期存在性

我们考虑具有无限深度周期性一维界面的重力水波系统，并严格证明 Dyachenko-Zakharov [16] 关于这些方程的近似可积性的猜想。更准确地说，我们证明了将水波方程严格简化为最高 4 阶的可积 Birkhoff 范式。因此，我们还获得了长期稳定性结果：初始大小为 ε 的平面界面的周期性扰动保持规律和小到订单时间${\epsilon }^{-\mathrm{3个}}$. 预计该时间尺度是最佳的。© 2022 作者。Communications on Pure and Applied Mathematics由 Wiley Periodicals LLC 出版。