### 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].

Language | English (US) |
---|---|

Pages | 417-429 |

Number of pages | 13 |

Journal | Bulletin of the Brazilian Mathematical Society |

Volume | 47 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bulletin of the Brazilian Mathematical Society*,

*47*(2), 417-429. DOI: 10.1007/s00574-016-0159-5

**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.

Research output: Contribution to journal › Article

*Bulletin of the Brazilian Mathematical Society*, vol. 47, no. 2, pp. 417-429. DOI: 10.1007/s00574-016-0159-5

}

TY - JOUR

T1 - On the spectral and modulational stability of periodic wavetrains for nonlinear Klein-Gordon equations

AU - Jones,Christopher K.R.T.

AU - Marangell,Robert

AU - Miller,Peter D.

AU - Plaza,Ramón G.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - 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].

AB - 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].

KW - modulation theory

KW - nonlinear Klein-Gordon equation

KW - periodic wavetrains

KW - spectral stability

UR - http://www.scopus.com/inward/record.url?scp=84976428086&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976428086&partnerID=8YFLogxK

U2 - 10.1007/s00574-016-0159-5

DO - 10.1007/s00574-016-0159-5

M3 - Article

VL - 47

SP - 417

EP - 429

JO - Bulletin of the Brazilian Mathematical Society

T2 - Bulletin of the Brazilian Mathematical Society

JF - Bulletin of the Brazilian Mathematical Society

SN - 1678-7544

IS - 2

ER -