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A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth

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Abstract

In this article we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling gravity-capillary surface waves on water of infinite depth. The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable and the infinite-dimensional function space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. The imaginary part of the spectrum of the linearised Hamiltonian vector field consists of essential spectrum at the origin and a finite number of eigenvalues whose distribution is described geometrically. Periodic solutions to the spatial Hamiltonian system are detected using Iooss’s generalisation of the reversible Lyapunov centre theorem; these solutions correspond to doubly periodic solutions to the travelling water-wave problem. For a generic choice of the periodic domain there exist values of the physical parameters at which a doubly periodic wave with this domain exists.

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Acknowledgments

We would like to thank Mariana Haragus for many helpful discussions during the preparation of this article. Thanks are also due to Erik Wahlén, who pointed out that the fractional-order Sobolev spaces used in previous spatial dynamics methods for three-dimensional water waves can be replaced by the simpler spaces \(\mathcal X_0\) and \(\mathcal X_1\) introduced in Sect. 2.2.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University, Loughborough, Leics, LE11 3TU, UK

    G. S. Bagri & M. D. Groves

  2. FR 6.1-Mathematik, Universität des Saarlandes, Postfach 15 11 50, 66041, Saarbrücken, Germany

    M. D. Groves

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  1. G. S. Bagri
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  2. M. D. Groves
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Correspondence to M. D. Groves.

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In memory of Klaus Kirchgässner.

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Bagri, G.S., Groves, M.D. A Spatial Dynamics Theory for Doubly Periodic Travelling Gravity-Capillary Surface Waves on Water of Infinite Depth. J Dyn Diff Equat 27, 343–370 (2015). https://doi.org/10.1007/s10884-013-9346-x

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  • DOI: https://doi.org/10.1007/s10884-013-9346-x

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