[Solutions homoclines de systèmes réversibles en présence d'un spectre essentiel.]
On étudie les bifurcations d'une classe de systèmes dynamiques réversibles de dimension infinie. Ces systèmes possèdent une famille de solutions stationnaires près de l'origine. On suppose que l'opérateur linéarisé à l'origine a un spectre essentiel sur l'axe réel et une valeur propre simple en 0. Une paire de valeurs propres imaginaires pour les valeurs du paramètre se rencontrent à l'origine pour et disparaissent pour . On donne ici des hypothèses sur et sur le terme non linéaire qui précisent la situation. Avec ces hypothèses on montre l'existence d'une famille de solutions homoclines aux solutions d'équilibre près de l'origine. Ce résultat s'applique à la recherche d'ondes solitaires dans des couches superposées de fluides parfaits, la couche inférieure étant de profondeur infinie.
In this Note we consider bifurcations of a class of infinite dimensional reversible dynamical systems. These systems possess a family of equilibrium solutions near the origin. We also assume that the linearized operator at the origin has an essential spectrum filling the entire real line, in addition to a simple eigenvalue at 0. Moreover, for parameter values there is a pair of imaginary eigenvalues which meet in 0 for , and which disappear for . We give assumptions on and on the non-linear term which describe this situation. These assumptions are sufficient to prove the existence of a family of solutions homoclinic to the equilibrium solutions near the origin. The result of this Note applies when we look for solitary waves in superposed layers of perfect fluids, the bottom one being infinitely deep.
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@article{CRMATH_2004__339_8_591_0,
author = {Barrandon, Matthieu},
title = {Homoclinic solutions of reversible systems possessing an essential spectrum},
journal = {Comptes Rendus. Math\'ematique},
pages = {591--596},
publisher = {Elsevier},
volume = {339},
number = {8},
year = {2004},
doi = {10.1016/j.crma.2004.07.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.07.001/}
}
TY - JOUR AU - Barrandon, Matthieu TI - Homoclinic solutions of reversible systems possessing an essential spectrum JO - Comptes Rendus. Mathématique PY - 2004 SP - 591 EP - 596 VL - 339 IS - 8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.07.001/ DO - 10.1016/j.crma.2004.07.001 LA - en ID - CRMATH_2004__339_8_591_0 ER -
%0 Journal Article %A Barrandon, Matthieu %T Homoclinic solutions of reversible systems possessing an essential spectrum %J Comptes Rendus. Mathématique %D 2004 %P 591-596 %V 339 %N 8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.07.001/ %R 10.1016/j.crma.2004.07.001 %G en %F CRMATH_2004__339_8_591_0
Barrandon, Matthieu. Homoclinic solutions of reversible systems possessing an essential spectrum. Comptes Rendus. Mathématique, Tome 339 (2004) no. 8, pp. 591-596. doi : 10.1016/j.crma.2004.07.001. http://www.numdam.org/articles/10.1016/j.crma.2004.07.001/
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