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Birkhoff Normal form for Gravity Water Waves

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Abstract

We consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff normal form (Berti et al. in Birkhoff normal form and long-time existence for periodic gravity Water Waves. arXiv:1810.11549, 2018). This confirms a conjecture of Zakharov–Dyachenko (Phys Lett A 190:144–148, 1994) based on the formal Birkhoff integrability of the water waves Hamiltonian truncated at degree four. As a consequence, we also obtain a long-time stability result: periodic perturbations of a flat interface that are of size \(\varepsilon \) in a sufficiently smooth Sobolev space lead to solutions that remain regular and small up to times of order \(\varepsilon ^{-3}\).

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Notes

  1. More in general, for a symbol \(a=a(x,\xi )\), \(x\in {\mathbb {T}},\xi \in {\mathbb {R}}\), and \(u \in L^2({\mathbb {T}})\), we set

    $$\begin{aligned} T_{a(x,\xi )}u := {Op^{\mathrm {BW}}}(a)u := \frac{1}{\sqrt{2\pi }}\sum _{k\in {\mathbb {Z}}} \left( \sum _{j\in {\mathbb {Z}}}{\widehat{a}}\left( k-j, \frac{k+j}{2}\right) \chi \left( \frac{k-j}{|k+j|}\right) \, {\widehat{u}}(j) \right) \frac{e^{\mathrm{i} k x}}{\sqrt{2\pi }} \end{aligned}$$
    (2.3)

    where \({\widehat{a}}\) denotes the Fourier transform in x and \(\chi \) is an even smooth cutoff function supported on \([- 10^{-2},10^{-2}]\).

  2. This can also be deduced using invariance properties of (1.1) such as the reversibility and preservation of the subspace of even functions.

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Authors and Affiliations

  1. SISSA, Trieste, Italy

    Massimiliano Berti

  2. SISSA, University of Nantes, Trieste, Italy

    Roberto Feola

  3. University of Toronto, Toronto, Canada

    Fabio Pusateri

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  1. Massimiliano Berti
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  2. Roberto Feola
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  3. Fabio Pusateri
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Correspondence to Fabio Pusateri.

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This research was supported by PRIN 2015 “Variational methods, with applications to problems in mathematical physics and geometry”. The third author was supported in part by a start-up grant from the University of Toronto and NSERC grant RGPIN-2018-06487.

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Berti, M., Feola, R. & Pusateri, F. Birkhoff Normal form for Gravity Water Waves. Water Waves 3, 117–126 (2021). https://doi.org/10.1007/s42286-020-00024-y

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