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On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type

Ehrnstrom, Mats ; Groves, Mark D. and Wahlén, Erik LU (2012) In Nonlinearity 25(10). p.2903-2936
Abstract
We consider a class of pseudodifferential evolution equations of the form u(t) + (n(u) + Lu)(x) = 0, in which L is a linear smoothing operator and n is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimization principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.
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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Nonlinearity
volume
25
issue
10
pages
2903 - 2936
publisher
London Mathematical Society / IOP Science
external identifiers
  • wos:000309112600006
  • scopus:84866314268
ISSN
0951-7715
DOI
10.1088/0951-7715/25/10/2903
language
English
LU publication?
yes
id
fc233e8b-df29-4444-b076-5df94d767338 (old id 3190048)
date added to LUP
2016-04-01 10:56:21
date last changed
2022-04-04 22:36:15
@article{fc233e8b-df29-4444-b076-5df94d767338,
  abstract     = {{We consider a class of pseudodifferential evolution equations of the form u(t) + (n(u) + Lu)(x) = 0, in which L is a linear smoothing operator and n is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimization principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.}},
  author       = {{Ehrnstrom, Mats and Groves, Mark D. and Wahlén, Erik}},
  issn         = {{0951-7715}},
  language     = {{eng}},
  number       = {{10}},
  pages        = {{2903--2936}},
  publisher    = {{London Mathematical Society / IOP Science}},
  series       = {{Nonlinearity}},
  title        = {{On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type}},
  url          = {{http://dx.doi.org/10.1088/0951-7715/25/10/2903}},
  doi          = {{10.1088/0951-7715/25/10/2903}},
  volume       = {{25}},
  year         = {{2012}},
}