Fully localised three-dimensional gravity-capillary solitary waves on water of infinite depth

TL;DR

This paper proves the existence of fully localized three-dimensional gravity-capillary solitary waves in infinite-depth water by reducing the problem to a perturbed nonlinear Schrödinger equation and applying the implicit function theorem.

Contribution

It extends the existence theory of localized solitary waves to infinite-depth water, building on previous finite-depth results, using a novel reduction to a nonlinear Schrödinger equation.

Findings

Existence of localized 3D gravity-capillary solitary waves in infinite-depth water.

Persistence of symmetric solutions under perturbations.

Reduction of the water wave problem to a perturbed nonlinear Schrödinger equation.

Abstract

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence for water of finite depth has recently been established, and in this article we present an existence theory for water of infinite depth. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schr\"{o}dinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable version of the implicit-function theorem shows that they persist under perturbations.