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This paper has been published in Physica D. To download a preprint of this paper just click here.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1206131. Any opinions, findings and conclusions or recomendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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Dynamical Hamiltonian-Hopf Instabilities of Periodic Traveling Waves in Klein-Gordon Equations

Robert Marangell and Peter D. Miller

School of Mathematics and Statistics F07, University of Sydney
Department of Mathematics, University of Michigan, Ann Arbor

Abstract:

We study the unstable spectrum close to the imaginary axis for the linearization of the nonlinear Klein-Gordon equation about a periodic traveling wave in a co-moving frame. We define dynamical Hamiltonian-Hopf instabilities as points in the stable spectrum that are accumulation points for unstable spectrum, and show how they can be determined from the knowledge of the discriminant of Hill's equation for an associated periodic potential. This result allows us to give simple criteria for the existence of dynamical Hamiltonian-Hopf instabilities in terms of instability indices previously shown to be useful in stability analysis of periodic traveling waves. We also discuss how these methods can be applied to more general nonlinear wave equations.