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Stability of some stationary solutions for the forced KdV equation

Camassa, R. and Wu, T. Yao-tsu (1991) Stability of some stationary solutions for the forced KdV equation. Physica D, 51 (1-3). pp. 295-307. ISSN 0167-2789. doi:10.1016/0167-2789(91)90240-a. https://resolver.caltech.edu/CaltechAUTHORS:20200306-133728214

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Abstract

The forced KdV (fKdV) equation has been established by recent studies as a simple mathematical model capable of describing the physics of a shallow layer of fluid in response to external forcing. For a particular one-parameter family of forcings characterized by a wave amplitude parameter for a certain supercritical forcing distribution, the exact stationary solutions are known. We study the stability of these solutions as the parameter varies. The linear stability analysis is first carried out by exploring the structure of the spectrum of the associated eigenvalue problem using a perturbation technique applied to a neighbourhood of isolated parameter values, where eigenfunctions corresponding to eigenvalue zero can be expressed in closed form. The results identify a set of intervals in the parameter space corresponding to different types of manifestation of instability. In the region of the parameter space where the linear stability analysis fails to provide an answer, we discuss a nonlinear analysis that provides a sufficient condition for stability.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/0167-2789(91)90240-aDOIArticle
Additional Information: © 1991 Published by Elsevier. We are indebted to George Yates for helpful discussions. This work was jointly sponsored by DOE Contract W-7405-ENG-36, AFOSRISSA 900024, ONR Contract N00014-89-J1971 and NSF Grant 4 DMS-890 1440. Numerical computations were performed at the San Diego Supercomputer Center.
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)W-7405-ENG-36
Air Force Office of Scientific Research (AFOSR)900024
Office of Naval Research (ONR)N00014-89-J1971
NSF4 DMS-890 1440
Issue or Number:1-3
DOI:10.1016/0167-2789(91)90240-a
Record Number:CaltechAUTHORS:20200306-133728214
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200306-133728214
Official Citation:R. Camassa, T.Yao-tsu Wu, Stability of some stationary solutions for the forced KdV equation, Physica D: Nonlinear Phenomena, Volume 51, Issues 1–3, 1991, Pages 295-307, ISSN 0167-2789, https://doi.org/10.1016/0167-2789(91)90240-A.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101748
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:06 Mar 2020 23:13
Last Modified:16 Nov 2021 18:05

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