Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

International audienceWe consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin $% L_{\varepsilon }$ has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values $% \varepsilon <0$ there is a pair of imaginary eigenvalues which meet in 0 for $\varepsilon =0$, and which disappear for $\varepsilon >0$. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin - Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest)