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A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

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Abstract

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.

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Authors and Affiliations

  1. Fachrichtung 6.1 - Mathematik, Universität des Saarlandes, Postfach 151150, 66041, Saarbrücken, Germany

    M. D. Groves

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

    M. D. Groves

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

    S. M. Sun

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

    E. Wahlén

Authors
  1. M. D. Groves
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  2. S. M. Sun
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  3. E. Wahlén
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Correspondence to E. Wahlén.

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Communicated by V. Šverák

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Groves, M.D., Sun, S.M. & Wahlén, E. A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension. Arch Rational Mech Anal 220, 747–807 (2016). https://doi.org/10.1007/s00205-015-0941-3

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  • DOI: https://doi.org/10.1007/s00205-015-0941-3

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