Moyal Noncommutative Integrability and the Burgers-KdV Mapping

TL;DR

This paper explores the algebraic structures of Moyal noncommutative integrable models, establishing a mapping between noncommutative KdV and Burgers systems that could advance understanding of noncommutative 2D integrability.

Contribution

It introduces a novel mapping between noncommutative KdV and Burgers systems using Moyal momentum algebra, revealing new constraints and insights into NC integrability.

Findings

Mapped NC KdV and Burgers systems via Moyal algebra

Identified a linearizing constraint for the NC Burgers system

Connected the constraint to conformal current conservation

Abstract

The Moyal momentum algebra is, once again, used to discuss some important aspects of NC integrable models and 2d conformal field theories. Among the results presented, we setup algebraic structures and makes useful convention notations leading to extract non trivial properties of the Moyal momentum algebra. We study also the Lax pair building mechanism for particular examples namely, the noncommutative KdV and Burgers systems. We show in a crucial step that these two systems are mapped to each others through a crucial mapping. This makes a strong constraint on the NC Burgers system which corresponds to linearizing its associated differential equation. From the conformal field theory point of view, this constraint equation is nothing but the analogue of the conservation law of the conformal current. We believe that this mapping might help to bring new insights towards understanding the integrability of noncommutative 2d-systems.