Noncommutative Burgers Equation

TL;DR

This paper introduces a noncommutative version of the Burgers equation, demonstrating its integrability, linearization, exact solutions, and connection to noncommutative Yang-Mills equations, expanding the understanding of noncommutative integrable systems.

Contribution

It develops a noncommutative Burgers equation with Lax representation, a Cole-Hopf transformation, and links to noncommutative Yang-Mills theory, providing new insights into noncommutative integrability.

Findings

The noncommutative Burgers equation is completely integrable.

A noncommutative Cole-Hopf transformation linearizes the equation.

Exact solutions and a hierarchy are constructed.

Abstract

We present a noncommutative version of the Burgers equation which possesses the Lax representation and discuss the integrability in detail. We find a noncommutative version of the Cole-Hopf transformation and succeed in the linearization of it. The linearized equation is the (noncommutative) diffusion equation and exactly solved. We also discuss the properties of some exact solutions. The result shows that the noncommutative Burgers equation is completely integrable even though it contains infinite number of time derivatives. Furthermore, we derive the noncommutative Burgers equation from the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, we present a noncommutative version of the Burgers hierarchy by both the Lax-pair generating technique and the Sato's approach.