A variational reduction and the existence of a fully-localised solitary wave for the three-dimensional water-wave problem with weak surface tension

TL;DR

This paper establishes the existence of fully localized solitary waves in the three-dimensional water-wave problem with weak surface tension, using a variational reduction to a perturbed Davey-Stewartson equation.

Contribution

It provides the first rigorous existence proof for localized waves in the physically relevant weak surface tension regime, extending previous results for strong surface tension.

Findings

Existence of fully localized solitary waves for 0<β<1/3.

Reduction of the water-wave problem to a variational problem related to Davey-Stewartson.

Identification of a nontrivial critical point via constrained minimization.

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 has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.