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

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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.


Enheter & grupper
Externa organisationer
  • Loughborough University
  • Virginia Polytechnic Institute and State University
  • Saarland University

Ämnesklassifikation (UKÄ) – OBLIGATORISK

  • Strömningsmekanik och akustik
  • Matematisk analys
Sidor (från-till)747-807
Antal sidor61
TidskriftArchive for Rational Mechanics and Analysis
Utgåva nummer2
StatusPublished - 2016 maj 1
Peer review utfördJa