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

Abstract

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lɛ (where ɛ is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line, with in addition a simple eigenvalue in 0. Moreover, for ɛ<0, there is a pair of imaginary eigenvalues which meet in 0 when ɛ=0 and which disappear in the essential spectrum for ɛ>0. For ɛ>0 small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T0,∞) possibly except a discrete set of isolated points.