Multi-field representations of KP hierarchies and multi-matrix models

TL;DR

This paper explores multi-field representations of KP hierarchies derived from multi-matrix models, introduces new integrable hierarchies through Dirac reduction, and connects them to known hierarchies like KdV and Boussinesque.

Contribution

It presents a systematic reduction method for multi-field KP hierarchies, leading to new integrable systems with explicit Lax pair formulations.

Findings

New integrable hierarchies via Dirac reduction

Explicit Lax pair representations for reduced systems

Connections to KdV and Boussinesque hierarchies

Abstract

We discuss the integrable hierarchies that appear in multi--matrix models. They can be envisaged as multi--field representations of the KP hierarchy. We then study the possible reductions of this systems via the Dirac reduction method by suppressing successively one by one part of the fields. We find in this way new integrable hierarchies, of which we are able to write the Lax pair representations by means of suitable Drinfeld--Sokolov linear systems. At the bottom of each reduction procedure we find an $N$--th KdV hierarchy. We discuss in detail the case which leads to the KdV hierarchy and to the Boussinesque hierarchy, as well as the general case in the dispersionless limit.