Gravity solitary waves with polynomial decay to exponentially small ripples at infinity E. Lombardi?, G.Iooss† ?,† INLN, UMR CNRS-UNSA 6618 1361 route des Lucioles, 06560 Valbonne, France † Institut Universitaire de france Abstract In this paper, we study the travelling gravity waves of velocity c 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 den- sities ? = ?2 = 1 ? ?(1 ? ?) near 0+, the existence of periodic travelling waves of arbitrary small amplitude and the existence of generalized soli- tary waves with ripples at infinity of size larger than ? 52 and polynomial decay rate were established in [7]. In this paper we improve this former result by showing the existence of generalized solitary waves with expo- nentially small ripples at infinity (of order O(e? c? )). We conjecture the non existence of true solitary waves in this case. The proof is based on a spatial dynamical formulation of the problem combined with a study of the analytic continuation of the solutions in the complex field which enables one to obtain exponentially small upper bounds of the oscillatory integrals giving the size of the oscillations at infinity. 1
- travelling waves
- waves has
- eigenvalues ±i?
- exponentially small
- waves asymptotic
- periodic travelling
- small quantity
- purely imaginary
- generalized solitary