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arXiv · 2507.14264

A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations

Darryl D. Holm·Maneesh Kumar Singh·Oliver D. Street
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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.

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BibTeXRIS

Darryl D. Holm, Maneesh Kumar Singh, Oliver D. Street. 2025-07-18. A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations. https://arxiv.org/abs/2507.14264

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