Exponentially Small Estimate for the Amplitude of Capillary Ripples of a Generalized Solitary Wave

A recently discovered phenomenon of capillary-gravity waves on water is a solitary wave of elevation plus small-amplitude ripples at infinity. It has been conjectured by Beale [Comm. Pure Appl. Math.44 (1991), 211-257] that the amplitude of the ripples should be exponentially small beyond all orders. In this paper we prove this conjecture.