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

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

Research output: Contribution to journalArticle

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].

LanguageEnglish (US)
Pages417-429
Number of pages13
JournalBulletin of the Brazilian Mathematical Society
Volume47
Issue number2
DOIs
StatePublished - Jun 1 2016

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Nonlinear Klein-Gordon Equation
Modulational Instability
Spectral Analysis
Stability Analysis
Modulation
Propagation
Perturbation
Imply

Keywords

  • modulation theory
  • nonlinear Klein-Gordon equation
  • periodic wavetrains
  • spectral stability

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations. / Jones, Christopher K.R.T.; Marangell, Robert; Miller, Peter D.; Plaza, Ramón G.

In: Bulletin of the Brazilian Mathematical Society, Vol. 47, No. 2, 01.06.2016, p. 417-429.

Research output: Contribution to journalArticle

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