Gravity solitary waves with polynomial decay to exponentially small ripples at infinity

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 densities ρ = ρ2/ρ1 and λ=gh/c^2. For ε = 1 − λ(1 − ρ) near 0 + , the existence of periodic travelling waves of arbitrary small amplitude and the existence of generalized solitary waves with ripples at infinity of size larger than ε^{ 5/2 } 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 exponentially small ripples at infinity (of order O(e^{ − c/ε})). 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.
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E Lombardi, Gérard Iooss. Gravity solitary waves with polynomial decay to exponentially small ripples at infinity. Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, Elsevier, 2003, 20 (4), pp.36. 〈10.1016/S0294-1449(02)00023-9〉. 〈hal-01271079〉

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