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Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity

Groves, M. D. and Wahlén, Erik LU (2015) In Proceedings of the Royal Society of Edinburgh. Section A 145(4). p.791-883
Abstract
We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy H subject to the constraint I = 2 mu, where I is the wave momentum and 0 < mu << 1. Since H and I are both conserved quantities, a standard argument asserts the stability of the set D-mu of minimizers: solutions starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg-de Vries... (More)
We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy H subject to the constraint I = 2 mu, where I is the wave momentum and 0 < mu << 1. Since H and I are both conserved quantities, a standard argument asserts the stability of the set D-mu of minimizers: solutions starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrodinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as mu down arrow 0. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
water waves, solitary waves, vorticity, calculus of variations
in
Proceedings of the Royal Society of Edinburgh. Section A
volume
145
issue
4
pages
791 - 883
publisher
Royal Society of Edinburgh
external identifiers
  • wos:000358440200008
  • scopus:84946185786
ISSN
0308-2105
DOI
10.1017/S0308210515000116
language
English
LU publication?
yes
id
9ce6fc9c-51b7-4326-9b9d-1afcc260ee48 (old id 7767751)
date added to LUP
2015-09-09 14:05:36
date last changed
2017-02-09 11:55:32
@article{9ce6fc9c-51b7-4326-9b9d-1afcc260ee48,
  abstract     = {We present an existence and stability theory for gravity-capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy H subject to the constraint I = 2 mu, where I is the wave momentum and 0 &lt; mu &lt;&lt; 1. Since H and I are both conserved quantities, a standard argument asserts the stability of the set D-mu of minimizers: solutions starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrodinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as mu down arrow 0.},
  author       = {Groves, M. D. and Wahlén, Erik},
  issn         = {0308-2105},
  keyword      = {water waves,solitary waves,vorticity,calculus of variations},
  language     = {eng},
  number       = {4},
  pages        = {791--883},
  publisher    = {Royal Society of Edinburgh},
  series       = {Proceedings of the Royal Society of Edinburgh. Section A},
  title        = {Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity},
  url          = {http://dx.doi.org/10.1017/S0308210515000116},
  volume       = {145},
  year         = {2015},
}