On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

Abstract

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy epsilon subject to the constraint I = root 2 mu, where I is the wave momentum and 0 < mu << 1. Since epsilon and I are both conserved quantities a standard argument asserts the stability of the set D-mu of minimisers: solutions starting near D-mu remain close to D-mu in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schrodinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mu down arrow 0.