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

Groves, M. D.; Sun, S. M. and Wahlén, E. LU (2016) In Archive for Rational Mechanics and Analysis 220(2). p.747-807
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... (More)

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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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Archive for Rational Mechanics and Analysis
volume
220
issue
2
pages
61 pages
publisher
Springer
external identifiers
  • scopus:84958876005
  • wos:000379423900007
ISSN
0003-9527
DOI
10.1007/s00205-015-0941-3
language
English
LU publication?
yes
id
76e27751-7ca3-43f7-a2a4-951eabb3970b
alternative location
https://arxiv.org/abs/1411.2475
date added to LUP
2016-05-10 09:30:01
date last changed
2017-04-16 04:37:27
@article{76e27751-7ca3-43f7-a2a4-951eabb3970b,
  abstract     = {<p>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.</p>},
  author       = {Groves, M. D. and Sun, S. M. and Wahlén, E.},
  issn         = {0003-9527},
  language     = {eng},
  month        = {05},
  number       = {2},
  pages        = {747--807},
  publisher    = {Springer},
  series       = {Archive for Rational Mechanics and Analysis},
  title        = {A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension},
  url          = {http://dx.doi.org/10.1007/s00205-015-0941-3},
  volume       = {220},
  year         = {2016},
}