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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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## Spectral and Modulational Stability of Periodic Wavetrains for the Nonlinear Klein-Gordon Equation

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

### Abstract:

This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon equation $$u_{tt}−u_{xx}+V′(u)=0$$, where $$u$$ is a scalar-valued function of $$x$$ and $$t$$, and the potential $$V(u)$$ is of class $$C^2$$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and present new spectral stability results for periodic traveling waves, and to make a solid connection between these results and predictions of the (formal) modulation theory, which has been developed by others but which we review for completeness.