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Perspectives on the Formation of Peakons in the Stochastic Camassa-Holm Equation

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arxiv 1910.03018 v3 pith:C2ANVRUO submitted 2019-10-04 math.NA cs.NA

Thomas M. Bendall , Colin J. Cotter , Darryl D. Holm This is my paper · ORCID
classification math.NAcs.NA
keywords peakonsequationstochasticbreakingcamassa-holmdiscretisationformationsolutions
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A famous feature of the Camassa-Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite element discretisation for it, which we use to explore the formation of peakons. Our simulations using this discretisation reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

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  1. Lagrangian averaged stochastic advection by Lie transport for fluids

    math-ph 2019-08 conditional novelty 5.0 of 10

    The paper formulates LA SALT stochastic fluid equations whose mean field satisfies a closed Lie-Laplacian Navier-Stokes equation, and establishes well-posedness and fluctuation variance dynamics.

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