Towards Noncommutative Integrable Equations

TL;DR

This paper extends integrable equations with Lax representations to noncommutative spaces, constructing hierarchies and demonstrating the complete integrability of a noncommutative Burgers equation.

Contribution

It develops a noncommutative Sato theory framework and derives new hierarchy equations, revealing hidden symmetries and integrability properties.

Findings

Derived noncommutative hierarchy equations

Established noncommutative Burgers equation as integrable

Suggested existence of infinite-dimensional symmetries

Abstract

We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has revealed essential aspects of the integrability of commutative soliton equations and the noncommutative extension is worth studying. We succeed in deriving various noncommutative hierarchy equations in the framework of the Sato theory, which is brand-new. The existence of the hierarchy would suggest a hidden infinite-dimensional symmetry in the noncommutative Lax equations. We finally show that a noncommutative version of Burgers equation is completely integrable because it is linearizable via noncommutative Cole-Hopf transformation. These results are expected to lead to the completion of the noncommutative Sato theory.