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A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

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Abstract

Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number \({\beta}\) greater than \({\frac{1}{3}}\)) has recently been given. In this article we present an existence theory for the physically more realistic case \({0 < \beta < \frac{1}{3}}\). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

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Authors and Affiliations

  1. Section de mathématiques, Station 8, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Boris Buffoni

  2. Fachrichtung Mathematik, Universität des Saarlandes, Postfach 151150, 66041, Saarbrücken, Germany

    Mark D. Groves

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

    Mark D. Groves

  4. Centre for Mathematical Sciences, Lund University, P.O. Box 118, 22100, Lund, Sweden

    Erik Wahlén

Authors
  1. Boris Buffoni
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  2. Mark D. Groves
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  3. Erik Wahlén
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Corresponding author

Correspondence to Boris Buffoni.

Additional information

Communicated by P. Rabinowitz

M. D. Groveswould like to thank theKnut andAliceWallenberg Foundation for funding a visiting professorship at Lund University during which this paper was prepared. E.Wahlén was supported by the Swedish Research Council (Grant No. 621-2012-3753).

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Buffoni, B., Groves, M.D. & Wahlén, E. A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension. Arch Rational Mech Anal 228, 773–820 (2018). https://doi.org/10.1007/s00205-017-1205-1

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  • DOI: https://doi.org/10.1007/s00205-017-1205-1

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