We study the existence of travelling breathers in Klein–Gordon chains, which describe nonlinear oscillators linearly coupled in a local anharmonic potential. In this work, we consider a case when the period of the breather and the inverse of the velocity are commensurable. In a neighborhood of critical values of velocity and coupling, we show by a center manifold reduction that the infinite-dimensional problem can be locally reduced to a eight-dimensional reversible differential equation.
Dans cette Note, on étudie l'existence de travelling breathers pour des chaînes de Klein–Gordon, qui décrivent des oscillateurs non linéaires linéairement couplés entre eux et plongés dans un potentiel anharmonique. Dans ce travail, on considère un cas où la période du breather et l'inverse de sa vitesse sont commensurables. Dans un voisinage de valeurs critiques pour le couplage et la vitesse, on montre, par réduction à une variété centrale, que le problème en dimension infinie se réduit localement à une équation différentielle réversible en dimension huit.
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@article{CRMATH_2004__338_8_661_0, author = {Sire, Yannick and James, Guillaume}, title = {Travelling breathers in {Klein{\textendash}Gordon} chains}, journal = {Comptes Rendus. Math\'ematique}, pages = {661--666}, publisher = {Elsevier}, volume = {338}, number = {8}, year = {2004}, doi = {10.1016/j.crma.2004.01.031}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2004.01.031/} }
TY - JOUR AU - Sire, Yannick AU - James, Guillaume TI - Travelling breathers in Klein–Gordon chains JO - Comptes Rendus. Mathématique PY - 2004 SP - 661 EP - 666 VL - 338 IS - 8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2004.01.031/ DO - 10.1016/j.crma.2004.01.031 LA - en ID - CRMATH_2004__338_8_661_0 ER -
%0 Journal Article %A Sire, Yannick %A James, Guillaume %T Travelling breathers in Klein–Gordon chains %J Comptes Rendus. Mathématique %D 2004 %P 661-666 %V 338 %N 8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2004.01.031/ %R 10.1016/j.crma.2004.01.031 %G en %F CRMATH_2004__338_8_661_0
Sire, Yannick; James, Guillaume. Travelling breathers in Klein–Gordon chains. Comptes Rendus. Mathématique, Volume 338 (2004) no. 8, pp. 661-666. doi : 10.1016/j.crma.2004.01.031. http://archive.numdam.org/articles/10.1016/j.crma.2004.01.031/
[1] Physica D, 119 (1998), pp. 34-46
[2] Physica D, 127 (1999), pp. 61-72
[3] Nonlinearity, 12 (1999), pp. 1601-1627
[4] Commun. Math. Phys., 161 (1994), pp. 391-418
[5] Nonlinearity, 13 (2000), pp. 849-866
[6] Topics in Bifurcation Theory and Applications, Adv. Ser. Nonlinear Dynam., vol. 3, World Sci. Publishing, 1992
[7] Commun. Math. Phys., 211 (2000), pp. 439-464
[8] J. Differential Equations, 102 (1993), pp. 62-88
[9] J. Nonlinear Sci., 13 (2003), pp. 27-63
[10] Phenomena beyond all orders and bifurcations of reversible homoclinic connections near higher resonances, Lecture Notes in Math., vol. 1741, Springer-Verlag, 2000
[11] Nonlinearity, 7 (1994), pp. 1623-1643
[12] Phys. Rev. B, 33 (1986), p. 2386
[13] Physica D, 138 (2000), pp. 267-281
[14] Center manifold theory in infinite dimensions, Dynam. Report (N.S.) (1992), pp. 125-163
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