Properties of Moyal-Lax Representation

TL;DR

This paper introduces the Moyal-Lax representation and momentum algebra, revealing their Hamiltonian structure, connection to non-commutativity, and role in deriving dispersionless integrable models' Hamiltonian structures.

Contribution

It systematically investigates the Moyal-Lax representation, linking non-commutativity to the system's central charge and deriving the second Hamiltonian structure for dispersionless models.

Findings

Moyal-Lax equation can be interpreted as a Hamiltonian equation.

Parameter of non-commutativity relates to the central charge.

Provides the second Hamiltonian structure for dispersionless integrable models.

Abstract

The Moyal-Lax representation and the Moyal momentum algebra are introduced and systematically investigated. It is shown that the Moyal-Lax equation can be interpreted as a Hamiltonian equation and can be derived from an action. We show that the parameter of non-commutativity, in this case, is related to the central charge of the second Hamiltonian structure of the system. The Moyal-Lax description leads in a natural manner to the dispersionless limit and provides the second Hamiltonian structure of dispersionless integrable models, which has been an open question for sometime.