Perspectives on the Formation of Peakons in the Stochastic Camassa-Holm Equation

TL;DR

This paper investigates how peakon solutions of the Camassa-Holm equation form under stochastic transport, demonstrating through finite element simulations that peakons can still develop despite randomness.

Contribution

It introduces a finite element discretisation for the stochastic Camassa-Holm equation and explores peakon formation mechanisms under stochastic influences.

Findings

Peakons can form with stochastic perturbations.

Peakons emerge via wave breaking and inflection point rise.

Finite element method effectively captures peakon dynamics.

Abstract

A famous feature of the Camassa-Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite element discretisation for it, which we use to explore the formation of peakons. Our simulations using this discretisation reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.