Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves

Frédéric Chardard

doi:10.1016/j.crma.2007.11.003

Dynamical Systems

Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves

Presented by: Gérard Iooss

Frédéric Chardard ¹

¹ CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue President Wilson, F-94230 Cachan, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 689-694.

Abstract
Résumé

A Maslov index for a solitary wave can be defined by approximating the solitary wave with periodic waves: when a sequence of periodic waves $ϕ^{α}$ converges to the solitary wave ϕ, then the sequence of Maslov indices converges and its limit can be used as a definition for the Maslov index of ϕ.

On peut définir l'indice de Maslov pour une onde solitaire en approchant l'onde solitaire par des ondes périodiques : lorsqu'une suite d'ondes périodiques $ϕ^{α}$ converge vers l'onde solitaire ϕ, alors sa limite peut-être utilisée comme définition de l'indice de Maslov de ϕ.

Received: 2007-07-12
Accepted: 2007-10-31
Published online: 2007-11-28

DOI: 10.1016/j.crma.2007.11.003

Author's affiliations:

Frédéric Chardard ¹

¹ CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue President Wilson, F-94230 Cachan, France

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     author = {Fr\'ed\'eric Chardard},
     title = {Maslov index for solitary waves obtained as a limit of the {Maslov} index for periodic waves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {689--694},
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     language = {en},
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Frédéric Chardard. Maslov index for solitary waves obtained as a limit of the Maslov index for periodic waves. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 689-694. doi : 10.1016/j.crma.2007.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.003/

References
Cited by

[1] J. Alexander; R. Gardner; C.K.R.T. Jones A topological invariant arising in the stability analysis of traveling waves, J. Reine Angew. Math., Volume 410 (1990), pp. 167-272

[2] V.I. Arnol'd Characteristic class entering in quantization conditions, Funktsional. Anal. i Prilozhen., Volume 1 (1967) no. 1, pp. 1-14

[3] A. Bose; C.K.R.T. Jones Stability of the in-phase travelling wave solution in a pair of coupled nerve fibers, Indiana Univ. Math. J., Volume 44 (1995) no. 1, pp. 189-220

[4] F. Chardard, F. Dias, T.J. Bridges, Computing the Maslov index of solitary waves, in preparation

[5] F. Chardard; F. Dias; T.J. Bridges Fast computation of the Maslov Index for hyperbolic linear systems with periodic coefficients, J. Phys. A: Math. Gen., Volume 39 (2006) no. 47, pp. 14545-14557

[6] M. Chugunova; D. Pelinovsky Count of eigenvalues in the generalized eigenvalue problem, 2006 (preprint, arXiv: pp. 1–30) | arXiv

[7] R.A. Gardner Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math., Volume 491 (1997), pp. 149-181

[8] C.K.R.T. Jones Instability of standing waves for non-linear Schrödinger-type equations, Ergodic Theory Dynam. Systems, Volume 8* (1988), pp. 119-138

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