Fully localised three-dimensional solitary water waves on Beltrami flows with strong surface tension

TL;DR

This paper develops an existence theory for fully localized three-dimensional solitary water waves with strong surface tension on Beltrami flows, reducing the problem to a perturbed KP-I equation and demonstrating the persistence of localized solutions.

Contribution

It introduces a novel existence framework for 3D solitary waves on Beltrami flows with strong surface tension, linking the problem to the KP-I equation and proving solution persistence.

Findings

Existence of fully localized 3D solitary water waves on Beltrami flows.

Reduction of the governing equations to a perturbed KP-I equation.

Persistence of localized solutions under perturbations.

Abstract

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. This paper presents an existence theory for such waves under the assumptions that the relative vorticity and velocity fields are parallel (`Beltrami flows'), that the free surface of the water takes the form $\{z=\eta(x,y)\}$ for some function $\eta: {\mathbb R}^2\rightarrow{\mathbb R}$, and that the influence of surface tension is sufficiently strong. The governing equations are formulated as a single equation for $\eta$, which is then reduced to a perturbation of the KP-I equation. This equation has recently been shown to have a family of nondegenerate localised solutions, and an application of a suitable variant of the implicit-function theorem shows that they persist under perturbations.