Variational Principles for Stochastic Soliton Dynamics

TL;DR

This paper introduces a variational approach to derive stochastic PDEs, demonstrates numerical simulations of stochastic peakon solutions in the Camassa-Holm equation, and explores how different stochastic perturbations affect peakon interactions.

Contribution

It presents a novel variational method for deriving stochastic PDEs and investigates the effects of different stochastic perturbations on soliton solutions.

Findings

Peakons persist under stochastic perturbations.

Different stochastic models lead to distinct peakon behaviors.

Some stochastic perturbations allow peakons to interpenetrate and exchange order.

Abstract

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-Poincar\'e 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.