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

TL;DR

This paper establishes the existence and stability of gravity-capillary solitary water waves with constant vorticity using a variational approach, and shows their convergence to classical model equations as wave amplitude diminishes.

Contribution

It introduces a variational framework for proving the existence and stability of solitary waves with vorticity, linking these solutions to classical models in the small amplitude limit.

Findings

Existence of minimizers for the wave energy under a momentum constraint.

Stability of the set of minimizers based on conserved quantities.

Convergence of the solutions to classical equations like KdV or nonlinear Schrödinger as amplitude decreases.

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 minimiser of the wave energy $\mathcal H$ subject to the constraint $\mathcal I=2\mu$, where $\mathcal I$ is the wave momentum and $0< \mu \ll 1$. Since $\mathcal H$ and $\mathcal I$ 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 described by solutions of a Korteweg-deVries equation (for strong surface tension) or a nonlinear Schr\"{o}dinger 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 $\mu \downarrow 0$