On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type
(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.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3190048
- author
- Ehrnstrom, Mats ; Groves, Mark D. and Wahlén, Erik LU
- organization
- publishing date
- 2012
- 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}}, }