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 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 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 $\mu \downarrow 0$

Corrected version. To appear in Proceedings of the Royal Society of Edinburgh: Section A

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