A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations

2025/07/18 by Holm, Darryl D., Singh, Maneesh Kumar, Street, Oliver D.
#70L10 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2507.14264

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.

Citations

• A Multiscale Camassa--Holm Equation
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Related

• Damping-Diffusion-Noise Interactions in the Stochastic Camassa-Holm Equation
• Perspectives on the Formation of Peakons in the Stochastic Camassa-Holm Equation
• Global existence of dissipative solutions to the Camassa–Holm equation with transport noise
• Modulation analysis of the stochastic Camassa-Holm equation with pure jump noise
• Casimir-dissipation stabilized stochastic rotating shallow water equations on the sphere