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A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Buffoni, Boris; Groves, Mark D. and Wahlén, Erik LU (2017) In Archive for Rational Mechanics and Analysis
Abstract

Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number (Formula presented.) greater than (Formula presented.)) has recently been given. In this article we present an existence theory for the physically more realistic case (Formula presented.). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle... (More)

Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number (Formula presented.) greater than (Formula presented.)) has recently been given. In this article we present an existence theory for the physically more realistic case (Formula presented.). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

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author
organization
publishing date
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Contribution to journal
publication status
epub
subject
in
Archive for Rational Mechanics and Analysis
pages
48 pages
publisher
Springer
external identifiers
  • scopus:85037642362
ISSN
0003-9527
DOI
10.1007/s00205-017-1205-1
language
English
LU publication?
yes
id
725bcd15-3578-4011-98e3-46584495c40b
date added to LUP
2017-12-21 10:05:54
date last changed
2018-01-07 12:29:37
@article{725bcd15-3578-4011-98e3-46584495c40b,
  abstract     = {<p>Fully localised solitary waves are travelling-wave solutions of the three- dimensional gravity–capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as ‘lumps’), and a mathematically rigorous existence theory for strong surface tension (Bond number (Formula presented.) greater than (Formula presented.)) has recently been given. In this article we present an existence theory for the physically more realistic case (Formula presented.). A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey–Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.</p>},
  author       = {Buffoni, Boris and Groves, Mark D. and Wahlén, Erik},
  issn         = {0003-9527},
  language     = {eng},
  month        = {12},
  pages        = {48},
  publisher    = {Springer},
  series       = {Archive for Rational Mechanics and Analysis},
  title        = {A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension},
  url          = {http://dx.doi.org/10.1007/s00205-017-1205-1},
  year         = {2017},
}