Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity

Abstract

We present an existence and stability theory for gravity–capillary solitary waves with constant vorticity on the surface of a body of water of finite depth. Exploiting a rotational version of the classical variational principle, we prove the existence of a minimizer of the wave energy 𝓗 subject to the constraint 𝓘 = 2µ, where 𝓘 is the wave momentum and 0 < µ ≪ 1. Since 𝓗 and 𝓘 are both conserved quantities, a standard argument asserts the stability of the set D_µ of minimizers: solutions starting near D_µ remain close to D_µ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are described by solutions of a Korteweg–de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the appropriate model equation as µ ↓ 0.