Perspectives on the Formation of Peakons in the Stochastic Camassa-Holm Equation
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The stochastic Camassa-Holm equation was derived in Holm and Tyranowski (2016) from the stochastic variational formulation of Holm (2015). Holm and Tyranowski (2016) also derived stochastic differential equations describing the evolution of the momenta and positions of peakon solutions. The work of Crisan and Holm (2018) then showed that in the stochastic Camassa-Holm equation peakons form with positive probabilities. This probability has not however yet been shown to be unity, unlike the deterministic case. We attempt to extend this discussion, by showing that peakons satisfying the stochastic differential equations presented in Holm and Tyranowski (2016) do indeed satisfy the stochastic Camassa-Holm equation, by writing the stochastic Camassa-Holm equation in a hydrodynamic form. Then we present a finite element discretisation of the stochastic Camassa-Holm equation, which we show numerically converges to the stochastic differential equations governing the evolution of peakons. We proceed to use this discretisation to numerically investigate the probability of peakons forming in the stochastic Camassa-Holm equation.
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