Dynamics of Gravity-Capillary Solitary Waves in Deep Water

TL;DR

This paper investigates the complex behavior of gravity-capillary solitary waves in deep water using numerical methods, simplifying the full potential flow equations to efficiently analyze wave dynamics, stability, and interactions.

Contribution

It introduces a cubic truncation of the Dirichlet-to-Neumann operator for efficient simulation of 3D solitary wave dynamics, including stability and interactions.

Findings

Good agreement with full equations for bifurcation and wave profiles

Observation of nonlinear focusing and quasi-elastic collisions

Identification of propagating localized breathers

Abstract

The dynamics of solitary gravity-capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time dependent solutions, we simplify the full potential flow problem by taking a cubic truncation of the scaled Dirichlet-to-Neumann operator for the normal velocity on the free surface. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain. Fully localised solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. The solitary wave branches are indexed by their finite energy at small amplitude, and the dynamics of the solitary waves is complex involving nonlinear focussing of wave packets, quasi-elastic collisions, and the generation of propagating, spatially localised, time-periodic structures (breathers).