A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension
(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.
(Less)
- author
- Groves, M. D. ; Sun, S. M. and Wahlén, E. LU
- organization
- publishing date
- 2016-05-01
- 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
- project
- Nonlinear Water Waves
- 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
- 2024-05-03 00:26:25
@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}},
doi = {{10.1007/s00205-015-0941-3}},
volume = {{220}},
year = {{2016}},
}