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On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations

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Abstract

In this contribution, we summarize recent results [8, 9] on the stability analysis of periodicwavetrains for the sine-Gordon and general nonlinearKlein-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 [12]. The connection between these two approaches is made through a modulational instability index [9], 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 rotationalwaves are spectrally stable. Our proof of this fact corrects a frequently cited one given by Scott [11].

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, U.S.A.

    Christopher K. R. T. Jones

  2. School of Mathematics and Statistics F07, University of Sydney, Sydney, NSW, 2006, Australia

    Robert Marangell

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, U.S.A.

    Peter D. Miller

  4. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México Apdo., Postal 20-726, C.P. 01000, México D.F., Mexico

    Ramón G. Plaza

Authors
  1. Christopher K. R. T. Jones
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  2. Robert Marangell
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  3. Peter D. Miller
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  4. Ramón G. Plaza
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Corresponding author

Correspondence to Ramón G. Plaza.

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Jones, C.K.R.T., Marangell, R., Miller, P.D. et al. On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations. Bull Braz Math Soc, New Series 47, 417–429 (2016). https://doi.org/10.1007/s00574-016-0159-5

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  • DOI: https://doi.org/10.1007/s00574-016-0159-5

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