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].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Alexander, R. A. Gardner and C. K. R. T. Jones. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410 (1990), 167–212.
J. C. Bronski and M. A. Johnson. The modulational instability for a generalized KdV equation. Arch. Rational Mech. Anal., 197(2) (2010), 357–400.
C. Chicone. Themonotonicity of the period function for planarHamiltonian vector fields. J. Differential Equations, 69(3) (1987), 310–321.
R. A. Gardner. On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. (9), 72(5) (1993), 415–439.
M. A. Johnson. On the stability of periodic solutions of the generalized Benjamin- Bona-Mahony equation. Phys. D, 239(19) (2010), 1892–1908.
M. A. Johnson and K. Zumbrun. Rigorous justification of the Whithammodulation equations for the generalized Korteweg-de Vries equation. Stud. Appl. Math., 125(1) (2010), 69–89.
M. A. Johnson, K. Zumbrun and J. C. Bronski. On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions. Phys. D, 239(23-24) (2010), 2057–2065.
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza. On the stability analysis of periodic sine-Gordon traveling waves. Phys. D, 251(1) (2013), 63–74.
C. K. R. T. Jones, R. Marangell, P. D. Miller and R. G. Plaza. Spectral and modulational stability of periodic wavetrains for the nonlinear Klein-Gordon equation. J. Differential Equations, 257(12) (2014), 4632–4703.
A. C. Scott. A nonlinearKlein-Gordon equation. Am. J. Phys., 37(1) (1969),52–61.
A. C. Scott. Waveform stabilityon a nonlinearKlein-Gordon equation. Proc. IEEE, 57(7) (1969), 1338–1339.
G. B. Whitham. Non-linear dispersive waves. Proc. Roy. Soc. Ser. A, 283 (1965), 238–261.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-016-0159-5