A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations

TL;DR

This paper introduces a novel energy-preserving stochastic perturbation of the Camassa-Holm equation, compares it with a Casimir-preserving approach, and studies the impact of noise on peakon formation through numerical simulations.

Contribution

It presents a new stochastic model that conserves energy, contrasting it with a Casimir-preserving model, and analyzes their effects on peakon dynamics.

Findings

Energy-preserving stochastic model maintains solution spread around deterministic peakons.

Casimir-preserving approach leads to more dramatic peakon propagation.

Numerical simulations illustrate different behaviors of the two stochastic models.

Abstract

We introduce a stochastic perturbation of the Camassa-Holm equation such that, unlike previous formulations, energy is conserved by the stochastic flow. We compare this to a complementary approach which preserves Casimirs of the Poisson bracket. Through an energy preserving numerical implementation of the model, we study the influence of noise on the well-known 'peakon' formation behaviour of the solution. The energy conserving stochastic approach generates an ensemble of solutions which are spread around the deterministic Camassa-Holm solution, whereas the Casimir conserving alternative develops peakons which may propagate away from the deterministic solution more dramatically.