Comptes Rendus
Mathematical Problems in Mechanics/Partial Differential Equations
Homoclinic solutions of reversible systems possessing an essential spectrum
Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 591-596.

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 Lɛ has an essential spectrum filling the entire real line, in addition to a simple eigenvalue at 0. Moreover, for parameter values ɛ<0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ=0, and which disappear for ɛ>0. We give assumptions on Lɛ 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.

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 Lɛ 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 ɛ<0 se rencontrent à l'origine pour ɛ=0 et disparaissent pour ɛ>0. On donne ici des hypothèses sur Lɛ 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.001
Matthieu Barrandon 1

1 INLN, UMR 6618 CNRS-UNSA, 1361, route des Lucioles, 06560 Valbonne, France
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Matthieu Barrandon. Homoclinic solutions of reversible systems possessing an essential spectrum. Comptes Rendus. Mathématique, Volume 339 (2004) no. 8, pp. 591-596. doi : 10.1016/j.crma.2004.07.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.001/

[1] C. Amick On the theory of internal waves of permanent form in fluids of great depth, Trans. Amer. Math. Soc., Volume 364 (1994), pp. 399-419

[2] C. Amick; J. Toland Uniqueness and related analytic properties for the Benjamin–Ono equation – a nonlinear Neumann problem in the plane, Acta Math., Volume 105 (1989), pp. 1-49

[3] M. Barrandon, Reversible bifurcation of homoclinic solutions in presence of an essential spectrum, Preprint INLN, 2003

[4] T.B. Benjamin Internal waves of permanent form in fluids of great depth, J. Fluid Mech., Volume 29 (1967), pp. 559-592

[5] F. Dias; G. Iooss Water-waves as a spatial dynamical system (S. Friedlander; D. Serre, eds.), Handbook of Mathematical Fluid Dynamics, vol. II, Elsevier, 2003, pp. 443-499

[6] G. Iooss; E. Lombardi; S.M. Sun Gravity traveling waves for two superposed fluid layers of infinite depth: a new type of bifurcation, Philos. Trans. Roy. Soc. London Ser. A, Volume 360 (2002), pp. 2245-2336

[7] H. Ono Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, Volume 39 (1975), pp. 1082-1091

[8] S.M. Sun Existence of solitary internal waves in a two-layer fluid of infinite depth, Nonlinear Anal., Volume 30 (1997), pp. 5481-5490

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