A variational approach to solitary gravity-capillary interfacial waves with infinite depth

Abstract

We present an existence and stability theory for gravity-capillary solitary waves on the top surface of and interface between two perfect fluids of different densities, the lower one being of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy $\mathcal{E}$ subject to the constraint $\mathcal{I}=2\mu$, where $\mathcal{I}$ is the wave momentum and $0< \mu < \mu_0$, where $\mu_0$ is chosen small enough for the validity of our calculations. Since $\mathcal{E}$ 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. The solitary waves which we construct are of small amplitude and are to leading order described by the cubic nonlinear Schr\"odinger equation. They exist in a parameter region in which the `slow' branch of the dispersion relation has a strict non-degenerate global minimum and the corresponding nonlinear Schr\"odinger equation is of focussing type. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of the model equation as $\mu \downarrow 0$.