A comparative numerical study of stochastic Hamiltonian Camassa–Holm equations

Abstract We introduce a stochastic perturbation of the Camassa–Holm (CH) 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 CH solution, whereas the Casimir conserving alternative develops peakons which may propagate away from the deterministic solution more dramatically.