Soliton Equations Extracted from the Noncommutative Zero-Curvature Equation

TL;DR

This paper derives various soliton equations from a 4-dimensional noncommutative zero-curvature equation, which generalizes classical integrable systems using Moyal brackets and extended variables.

Contribution

It introduces a noncommutative zero-curvature framework with additional variables, extending classical soliton equations through dimensional reduction.

Findings

Derivation of soliton equations from a 4D noncommutative zero-curvature equation

Extension of classical integrable systems using Moyal brackets

New framework for noncommutative soliton equations

Abstract

We investigate the equation where the commutation relation in 2-dimensional zero-curvature equation composed of the algebra-valued potentials is replaced by the Moyal bracket and the algebra-valued potentials are replaced by the non-algebra-valued ones with two more new variables. We call the 4-dimensional equation the noncommutative zero-curvature equation. We show that various soliton equations are derived by the dimensional reduction of the equation.