[Paper Review] The stochastic integrable AKNS hierarchy

Alexis Arnaudon|arXiv (Cornell University)|Nov 22, 2015

Nonlinear Waves and Solitons27 references3 citations

TL;DR

This paper derives a stochastic integrable AKNS hierarchy using geometrical methods based on Lagrangian reduction and stochastic zero-curvature relations, preserving integrability under temporal noise. The key contribution is a systematic framework yielding stochastic NLS, KdV, mKdV, and Camassa-Holm equations with soliton solutions via stochastic inverse scattering transform, including novel stochastic peakon dynamics for type II CH equations.

ABSTRACT

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.

Motivation & Objective

• To develop a systematic method for adding temporal noise to integrable PDEs while preserving their complete integrability.
• To extend the stochastic KdV equation of Wadati to the full AKNS hierarchy using geometrical and isospectral methods.
• To formulate a stochastic inverse scattering transform (IST) compatible with the stochastic zero-curvature relation.
• To derive and analyze stochastic Camassa-Holm equations, particularly those yielding stochastic peakon systems.
• To explore the physical and mathematical implications of stochastic deformations in integrable systems, especially in the context of soliton dynamics and conservation laws.

Proposed method

• Uses Lagrangian reduction by symmetry to derive the deterministic AKNS hierarchy, then applies stochastic deformation via the framework of [28] for fluid dynamics and mechanical systems.
• Implements a stochastic zero-curvature relation (ZCR) as the integrability condition, ensuring the stochastic system remains integrable.
• Applies the stochastic inverse scattering transform (IST) to construct one-soliton solutions, particularly for the stochastic NLS equation.
• Derives stochastic Camassa-Holm equations of type I, II, and III by extending the AKNS framework, with type II yielding a stochastic peakon system.
• Uses the Stratonovich calculus formulation with noise terms proportional to the flow of the same hierarchy, ensuring geometric consistency.
• Applies the method to reduce the stochastic CH equation to a stochastic Hamiltonian system of peakons, preserving the structure of the deterministic case.

Experimental results

Research questions

• RQ1 Can a stochastic deformation of the AKNS hierarchy be constructed such that integrability is preserved under temporal noise?
• RQ2 How does the stochastic inverse scattering transform differ from the deterministic version, and what types of solutions can it generate?
• RQ3 What are the dynamical properties of stochastic peakons in the context of the stochastic Camassa-Holm equation of type II?
• RQ4 Can the stochastic deformation method be generalized to other integrable hierarchies beyond AKNS?
• RQ5 What is the physical and mathematical significance of stochastic terms that are proportional to the flow of the same hierarchy?

Key findings

• The stochastic AKNS hierarchy is derived via Lagrangian reduction and stochastic deformation, preserving integrability through a stochastic zero-curvature relation.
• The stochastic inverse scattering transform successfully generates one-soliton solutions for the stochastic NLS equation, extending Wadati’s stochastic KdV result.
• A stochastic Camassa-Holm equation of type II yields a stochastic Hamiltonian system of peakons with dynamics governed by $ dq_i = u(q_i,t)(dt + \circ dW) $, $ dp_i = -p_i u(q_i,t)(dt + \circ dW) $.
• The stochastic peakon system preserves conserved quantities such as momentum $ P = p_1 + p_2 $ and energy $ H = \frac{1}{2}(p_1^2 + p_2^2) $, preventing peakon crossings.
• The stochastic CH equation of type II has a Hamiltonian formally given by $ h(p,q)dt = \frac{1}{2}\sum_{i,j} p_i p_j e^{-|q_i - q_j|/\alpha}(dt + \circ dW) $, with identical stochastic potential as the deterministic Hamiltonian.
• The method is general and suggests that the derived stochastic equations are the only possible integrable stochastic versions of NLS, KdV, and mKdV equations under the given framework.