On the Spectral and Orbital Stability of Spatially Periodic Stationary Solutions of Generalized Korteweg-de Vries Equations

  • Todd Kapitula
  • Bernard DeconinckEmail author
Part of the Fields Institute Communications book series (FIC, volume 75)


In this paper we generalize previous work on the spectral and orbital stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric operator \(\mathcal{J}\) is singular. We assume that \(\mathcal{J}\) restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higher-order pure power nonlinearities. We conclude by studying the transverse stability of these solutions when they are considered as planar solutions of the generalized KP-I equation.


Orbital Stability Krein Signature Spectral Stability Relative Equilibrium Local Well-posedness Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



BD acknowledges support from the National Science Foundation through grant NSF-DMS-0604546. TK gratefully acknowledges the support of the Jack and Lois Kuipers Applied Mathematics Endowment, a Calvin Research Fellowship, and the National Science Foundation under grant DMS-0806636. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGrand RapidsUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

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