The stochastic integrable AKNS hierarchy

We derive a stochastic AKNS hierarchy using geometrical methods. The integrability is shown via a stochastic zero curvature relation associated with a stochastic isospectral problem. We expose some of the stochastic integrable partial differential equations which extend the stochastic KdV equation discovered by M. Wadati in 1983 for all the AKNS flows. We also show how to find stochastic solitons from the stochastic evolution of the scattering data of the stochastic IST. We finally expose some properties of these equations and also briefly study a stochastic Camassa-Holm equation which reduces to a stochastic Hamiltonian system of peakons.

💡 Research Summary

The paper develops a systematic geometric framework for introducing temporal white‑noise perturbations into the entire AKNS integrable hierarchy while preserving its complete integrability. The authors begin by recalling the loop‑group formulation of the AKNS hierarchy: the configuration space is the loop group (LG) of smooth maps from the circle into the Lie group (G=SL(2)). By endowing this group with a central extension defined via the cocycle
(c(\xi,\eta)=\int \xi,\partial_x\eta,dx),
they obtain the extended loop algebra (\widehat{L\mathfrak g}=L\mathfrak g\oplus\mathbb R). This structure naturally accommodates a connection one‑form (M=\sum_i M^{(i)}da_i) on an infinite‑dimensional “time” space (\mathbb R^\infty). The zero‑curvature condition (dM+