Reversible bifurcation of homoclinic solutions in presence of an essential spectrum
by
Matthieu Barrandon
INLN, 1361 route des Lucioles, 06560 Valbonne, France

We consider bifurcations of a class of infinite dimensional reversible dynamical systems possessing an equilibrium solution at the origin and such that the linearized operator at the origin L_\epsilon has an essential spectrum filling the entire real line, in addition to a simple eigenvalue at 0. We also assume that for parameter values \epsilon < 0, there is in addition a pair of imaginary eigenvalues which meet in 0 for \epsilon = 0, and which disappear for \epsilon > 0.

We can find examples of this situation in the theory of water-waves.

We characterize here the structure of such linear operators via the structure of the resolvent. We then consider the bifurcation problem and give quite general assumptions under which there is a bifurcation of a symmetric homoclinic solution with an algebraic decay at infinity, approximated at main order by the Benjamin - Ono homoclinic.

Date received: April 30, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-02.