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Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg–de Vries Equation9
Johnson, Mathew A.
Johnson, Mathew A.
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Abstract
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 $p\geq1$.
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This is the published version, also available here: http://dx.doi.org/10.1137/090752249.
Date
2009-11-18
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Society for Industrial and Applied Mathematics
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Keywords
generalized Korteweg-de Vries equation, periodic waves, orbital stability
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.