Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation9
Issue Date
2009-11-18Author
Johnson, Mathew A.
Publisher
Society for Industrial and Applied Mathematics
Type
Article
Article Version
Scholarly/refereed, publisher version
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In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg–de Vries equation $u_t=u_{xxx}+f(u)_x$. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solutions. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case when $f(u)=u^2$ (the Korteweg–de Vries equation) and in neighborhoods of the homoclinic and equilibrium solutions if $f(u)=u^{p+1}$ for some $p\geq1$.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/090752249.
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Citation
Johnson, Mathew A. "Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation." SIAM J. Math. Anal., 41(5), 1921–1947. (27 pages). http://dx.doi.org/10.1137/090752249.
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