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

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Author

Summary, in English

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.

Department/s

Publishing year

2016-05-01

Language

English

Pages

747-807

Publication/Series

Archive for Rational Mechanics and Analysis

Volume

220

Issue

2

Document type

Journal article

Publisher

Springer

Topic

  • Fluid Mechanics and Acoustics
  • Mathematical Analysis

Status

Published

Project

  • Nonlinear Water Waves

Research group

  • Partial differential equations

ISBN/ISSN/Other

  • ISSN: 0003-9527