Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

Abstract

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

Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

“…Therefore, the properties on the resolvent and on the nonlinear term (see Sections 2.3 and 2.4) for the water-wave problems are written in the general frame of spatial dynamics. These properties correspond to weaker assumptions than the ones made in [3].…”

“…This bifurcation has been encountered in [3], in which the existence of a one parameter family of bifurcated homoclinic solutions of (1) approximated by the Benjamin-Ono solitary wave is proved. Because of the presence of the essential spectrum in this paper, the sole description of the spectrum is not sufficient to prove the existence of these solutions.…”

“…In presence of an essential spectrum on the real axis, the singularity of the resolvent in k = 0 is unknown. That's why a meticulous description of the resolvent is performed in [3]: some assumptions on the resolvent of L ε are given in order to describe the singularity in k = 0. It is checked in [3] that these assumptions are satisfied in both water-wave problems presented above.…”

“…That's why a meticulous description of the resolvent is performed in [3]: some assumptions on the resolvent of L ε are given in order to describe the singularity in k = 0. It is checked in [3] that these assumptions are satisfied in both water-wave problems presented above. Therefore, there exists a one parameter family of solitary waves for problems 1 and 2 (corresponding to the homoclinic solutions in the spatial dynamics formulation).…”

A Variational Approach to Solitary Gravity–Capillary Interfacial Waves with Infinite Depth

“…not assumed to be a minimum). There are also a number of papers dealing with solitary or generalised solitary waves (asymptotic to periodic solutions at spatial infinity) in the related settings where either one or both of the surface and interfacial tension vanishes (see [1,2,11,17,23,24] and references therein). The variational method presented in this paper does not work in those settings since it requires both surface tension and interfacial tension.…”