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Hamiltonian Studies on Counter-Propagating Water Waves

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Abstract

We use a Hamiltonian normal form approach to study the dynamics of the water wave problem in the small-amplitude long-wave regime (KdV regime). If \(\mu \) is the small parameter corresponding to the inverse of the wave length, we show that the normal form at order \(\mu ^5\) consists of two decoupled equations: one describing right going waves and the other describing left going waves. Each of these equations is integrable: it is a linear combination of the first three equations in the KdV hierarchy. At order \(\mu ^7\), we find nontrivial terms coupling the two counter-propagating waves.

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Notes

  1. These are the \(L^1\) based Sobolev spaces. See below for a precise definition.

  2. Actually, one could get a slightly more precise statement using the Beppo Levi spaces; here, for the sake of simplicity, I decided not to use them.

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Acknowledgements

Part of the material presented in this paper is the content of some lectures that I gave more than 10 years ago to prepare a visit by Walter Craig. I thank all the people who attended such lectures and contributed with their comments to improve the material, in particular Antonio Ponno with whom I had a lot of discussions on the subject. The discussions with Antonio Ponno were also the key to the understanding of the relevance of Kodama’s work in the present context. I also would like to thank Doug Wright and David Lannes who gave me some relevant feedback on higher order corrections to KdV and on some technical issues. Finally, I warmly thank the two referees of the paper whose comments allowed to greatly improve the paper.

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  1. Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Dario Bambusi

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Bambusi, D. Hamiltonian Studies on Counter-Propagating Water Waves. Water Waves 3, 49–83 (2021). https://doi.org/10.1007/s42286-020-00032-y

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