Towards Noncommutative Integrable Systems

TL;DR

This paper introduces a method to generate noncommutative integrable equations with Lax representations in (1+1) and (1+2) dimensions, linking them to noncommutative Yang-Mills equations and exploring their integrability and string theory connections.

Contribution

A novel method for deriving noncommutative integrable equations from reductions of noncommutative Yang-Mills equations, supporting the noncommutative Ward conjecture.

Findings

Generated noncommutative integrable equations with Lax pairs.

Established connection to noncommutative Yang-Mills equations.

Discussed implications for string theory.

Abstract

We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1) and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.