Gravity travelling waves for two superposed fluid layers one being of

Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation By Gerard Iooss1,2, Eric Lombardi2, Shu Ming Sun3 1 IUF, 2 INLN, UMR CNRS-UNSA 6618, 1361 route des Lucioles, 06560 Valbonne, France 3Math. Dept., Virginia Tech, Blacksburg VA 24061, USA In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential, and the dimensionless parameters are the ratio between densities ? = ?2/?1 and ? = gh/c2. We study special values of the parameters such that ?(1? ?) is near 1?, where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with in addition a double eigenvalue in 0, a pair of simple imaginary eigenvalues ±i? at a distance O(1) from 0, and for ?(1??) above 1, another pair of simple imaginary eigenvalues tending towards 0 as ?(1? ?) ? 1+.

• travelling waves


• eigenvalues ±i?


• infinite dimension


• gravity travelling waves


• waves asymptotic


• periodic travelling


• imaginary eigenvalues


• spectrum


• i? resonance