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Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem

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Abstract

A model equation derived by Kadomtsev & Petviashvili (Sov Phys Dokl 15:539–541, 1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties. The reduced functional is related to the functional associated with the Kadomtsev–Petviashvili equation, and a nontrivial critical point is detected using the direct methods of the calculus of variations.

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Authors and Affiliations

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

    M. D. Groves

  2. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    S.-M. Sun

Authors
  1. M. D. Groves
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  2. S.-M. Sun
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Correspondence to M. D. Groves.

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Communicated by A. Mielke

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Groves, M.D., Sun, SM. Fully Localised Solitary-Wave Solutions of the Three-Dimensional Gravity–Capillary Water-Wave Problem. Arch Rational Mech Anal 188, 1–91 (2008). https://doi.org/10.1007/s00205-007-0085-1

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  • DOI: https://doi.org/10.1007/s00205-007-0085-1

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