Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

TL;DR

This paper proves the existence of fully localized three-dimensional gravity-capillary water waves with strong surface tension effects, demonstrating their energetic stability through a variational approach, despite the lack of a global well-posedness theory.

Contribution

It establishes the existence and conditional energetic stability of fully localized 3D water waves with strong surface tension using a variational method.

Findings

Existence of small-amplitude localized solitary waves.

These waves are critical points of the energy with fixed momentum.

Stability of the set of minimizers is demonstrated.

Abstract

In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed. We prove the existence of a minimiser of the energy subject to the constraint that the momentum is fixed and small. The existence of a small-amplitude solitary wave is thus assured, and since the energy and momentum are both conserved quantities a standard argument may be used to establish the stability of the set of minimisers as a whole. `Stability' is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.