Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies

TL;DR

This paper extends the Sato theory to noncommutative spaces, establishing the existence of infinite conservation laws for various noncommutative integrable hierarchies including KP, KdV, and Boussinesq.

Contribution

It provides explicit conservation laws and representations for noncommutative Lax hierarchies, advancing the understanding of integrability in noncommutative geometry.

Findings

Existence of infinite conserved densities for noncommutative hierarchies.

Explicit formulas for conservation laws in terms of Lax operators.

Extension of classical integrable equations to noncommutative settings.

Abstract

We discuss commuting flows and conservation laws for Lax hierarchies on noncommutative spaces in the framework of the Sato theory. On commutative spaces, the Sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the Lax hierarchies in terms of pseudo-differential operators. Noncommutative extension of the Sato theory has been already studied by the author and Kouichi Toda, and the existence of various noncommutative Lax hierarchies are guaranteed. In the present paper, we present conservation laws for the noncommutative Lax hierarchies with both space-space and space-time noncommutativities and prove the existence of infinite number of conserved densities. We also give the explicit representations of them in terms of Lax operators. Our results include noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera, modified KdV equations and so on.