随机孤子动力学的变分原理
Variational Principles for Stochastic Soliton Dynamics
摘要：
我们开发了一种变分方法，推导随机偏微分方程的解决方案，遵循流的随机向量场。作为一个例子，在一个空间维度，我们数值模拟奇异解（峰值）的随机扰动Camassa-Holm（CH）方程使用这种方法。这些数值模拟结果表明，峰值孤子解的随机扰动CH方程持续存在，并提供了一个有趣的实验室调查的灵敏度和精度随机偏微分方程（SPDE）的有限维解决方案的随机性。特别地，讨论了由维纳噪声引起的峰子动力学的随机扰动的一些选择（正则哈密顿随机变形，或CH-SD）允许峰子在超越碰撞中相互渗透并交换实数线上的顺序，尽管这种行为不会发生在保留CH方程的Euler-Poincaré结构的其他随机扰动选择中（参数随机变形，或P-SD），它也不会发生的未扰动的确定性CH方程的峰解。讨论提出了有关科学的随机变形的有限维近似的演化PDE和所得到的解决方案的敏感性随机建模中所作的选择。
英文摘要：
We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations (SPDE). In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, or CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the CH equation (parametric stochastic deformations, or P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary PDE and the sensitivity of the resulting solutions to the choices made in stochastic modelling.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
