Extension of Noncommutative Soliton Hierarchies

TL;DR

This paper extends noncommutative soliton hierarchies by incorporating deformation parameters as additional coordinates, leading to larger hierarchies with commuting flows and connections to Seiberg-Witten maps.

Contribution

It introduces a method to extend soliton hierarchies to include deformation equations, proving properties for deformed AKNS, NLS, KdV, mKdV, and KP hierarchies.

Findings

Extended hierarchies maintain integrability

Deformation equations commute with soliton flows

Seiberg-Witten maps relate classical and deformed solutions

Abstract

A linear system, which generates a Moyal-deformed two-dimensional soliton equation as integrability condition, can be extended to a three-dimensional linear system, treating the deformation parameter as an additional coordinate. The supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter, the flow of which commutes with the flow of the deformed soliton equation. In this way, a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations. We prove the extended hierarchy properties for the deformed AKNS hierarchy, and specialize to the cases of deformed NLS, KdV and mKdV hierarchies. Corresponding results are also obtained for the deformed KP hierarchy. A deformation equation determines a kind of Seiberg-Witten map from classical solutions to solutions of the respective `noncommutative' deformed equation.