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This paper was published in the Bulletin of the Brazilian Mathematical Society, in the proceedings of the XV International Conference on Hyperbolic Problems: Theory, Numerics, Applications (HYP2014), July 28 - August 1, 2014, IMPA, Rio de Janeiro, Brazil.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-0807653 and 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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On the Spectral and Modulational Stability of Periodic Wavetrains for Nonlinear Klein-Gordon Equations

Christopher K. R. T. Jones, Robert Marangell, Peter D. Miller, and Ramón G. Plaza

Department of Mathematics, University of North Carolina, Chapel Hill
School of Mathematics and Statistics F07, University of Sydney
Department of Mathematics, University of Michigan, Ann Arbor
Departamento de Matemáticas y Mecánica, IIMAS-FENOMEC, Universidad Nacional Autónoma de México


In this contribution, we summarize recent results on the stability analysis of periodic wavetrains for the sine-Gordon and general nonlinear Klein-Gordon equations. Stability is considered both from the point of view of spectral analysis of the linearized problem and from the point of view of the formal modulation theory of Whitham. The connection between these two approaches is made through a modulational instability index, which arises from a detailed analysis of the Floquet spectrum of the linearized perturbation equation around the wave near the origin. We analyze waves of both subluminal and superluminal propagation velocities, as well as waves of both librational and rotational types. Our general results imply in particular that for the sine-Gordon case only subluminal rotational waves are spectrally stable. Our proof of this fact corrects a frequently cited one given by Scott.