Numerical study of the Amick-Schonbek system in 2D

TL;DR

This paper presents a numerical analysis of the 2D Amick-Schonbek Boussinesq system, providing evidence for the transverse stability of 1D solitary waves and exploring conditions leading to gradient catastrophe or stable solutions.

Contribution

It offers the first numerical investigation into the stability and dynamics of the 2D Amick-Schonbek system, highlighting the role of the non-cavitation condition.

Findings

Transverse stability of 1D solitary waves confirmed numerically.

Gradient catastrophe can occur without the non-cavitation condition.

Stable solutions persist for all times when the non-cavitation condition is satisfied.

Abstract

A numerical study of the 2D Amick-Schonbek Boussinesq system is presented. Numerical evidence is given for the transverse stability of the 1D solitary waves that are line solitary waves of the 2D equations. It is shown that initial data not satisfying the non-cavitation condition can lead to the formation of a gradient catastrophe in finite time. The numerical propagation of localised smooth initial data does not lead to the formation of stable structures localised in both spatial directions. For initial data satisfying the non-cavitation condition, smooth solutions appear to exist for all times.