Homoclinic solutions of reversible systems possessing an essential spectrum

Abstract

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 L ɛ 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 L ɛ 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. To cite this article: M. Barrandon, C. R. Acad. Sci. Paris, Ser. I 339 (2004).