Numerical study of fractional Camassa-Holm equations

TL;DR

This paper conducts a numerical investigation of fractional Camassa-Holm equations, focusing on solitary wave construction, stability analysis, long-term dynamics, and the emergence of dispersive shock waves.

Contribution

It introduces numerical methods for fractional Camassa-Holm equations and explores their solitary waves, stability, and shock wave phenomena, which are novel in this context.

Findings

Construction of smooth solitary waves

Analysis of their stability

Observation of dispersive shock waves

Abstract

A numerical study of fractional Camassa-Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localised initial data from the Schwartz class of rapidly decreasing functions. The appearence of dispersive shock waves is explored.