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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
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
2012-12-04 08:24:51
date last changed
2017-09-17 04:04:38
@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},
  volume       = {25},
  year         = {2012},
}