Soliton equations and the zero curvature condition in noncommutative geometry

TL;DR

This paper explores how classical soliton equations like KdV, KP, and Boussinesq can be formulated as zero curvature conditions within noncommutative geometry, providing new insights into their structure and transformations.

Contribution

It introduces a framework where soliton equations are derived from noncommutative differential calculi, extending classical integrable systems into noncommutative geometry.

Findings

Formulation of Burgers, KdV, and KP equations as zero curvature conditions

Interpretation of Cole-Hopf transformation as a gauge transformation

Derivation of Boussinesq calculus from KP framework

Abstract

Familiar nonlinear and in particular soliton equations arise as zero curvature conditions for GL(1,R) connections with noncommutative differential calculi. The Burgers equation is formulated in this way and the Cole-Hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework. The KdV, modified KdV equation and the Miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the KP and a modified KP equation. Furthermore, a differential calculus associated with the Boussinesq equation is derived from the KP calculus.