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.
翻译:卡马萨-荷尔姆方程式的一个著名特征是它接纳了被称为峰值的峰值索利顿溶液。我们在随机迁移的影响下调查了这个方程式。我们注意到峰值是方程的微弱溶液,因此为它提出了一个有限的元素分解,我们用来探索峰值的形成。我们使用这种离散的模拟显示,峰值仍然可以在有随机扰动的情况下形成。峰值可以通过波流破碎而出现,随着斜坡的垂直旋转,随着速度剖面的偏移点升至峰顶,波破碎也会同时出现。