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On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

Groves, M. D. and Wahlén, Erik LU (2011) In Journal of Mathematical Fluid Mechanics 13(4). p.593-627
Abstract
This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy epsilon subject to the constraint I = root 2 mu, where I is the wave momentum and 0 < mu << 1. Since epsilon and I are both conserved quantities a standard argument asserts the stability of the set D-mu of minimisers: 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 modelled as solutions of the nonlinear Schrodinger equation with cubic... (More)
This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy epsilon subject to the constraint I = root 2 mu, where I is the wave momentum and 0 < mu << 1. Since epsilon and I are both conserved quantities a standard argument asserts the stability of the set D-mu of minimisers: 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 modelled as solutions of the nonlinear Schrodinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mu down arrow 0. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Mathematical Fluid Mechanics
volume
13
issue
4
pages
593 - 627
publisher
Birkhaüser
external identifiers
  • wos:000300352500007
  • scopus:80855165477
ISSN
1422-6928
DOI
10.1007/s00021-010-0034-x
language
English
LU publication?
yes
id
1849dc9d-8056-40d1-b433-e7f3de6d851c (old id 2377716)
date added to LUP
2012-03-27 12:04:47
date last changed
2017-07-23 04:00:11
@article{1849dc9d-8056-40d1-b433-e7f3de6d851c,
  abstract     = {This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy epsilon subject to the constraint I = root 2 mu, where I is the wave momentum and 0 &lt; mu &lt;&lt; 1. Since epsilon and I are both conserved quantities a standard argument asserts the stability of the set D-mu of minimisers: 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 modelled as solutions of the nonlinear Schrodinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mu down arrow 0.},
  author       = {Groves, M. D. and Wahlén, Erik},
  issn         = {1422-6928},
  language     = {eng},
  number       = {4},
  pages        = {593--627},
  publisher    = {Birkhaüser},
  series       = {Journal of Mathematical Fluid Mechanics},
  title        = {On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water},
  url          = {http://dx.doi.org/10.1007/s00021-010-0034-x},
  volume       = {13},
  year         = {2011},
}