The (N,M)-th KdV hierarchy and the associated W algebra

TL;DR

This paper introduces the (N,M)-th KdV hierarchy, a new integrable system extending the KdV hierarchy with pseudo-differential terms, revealing connections to W algebras and multi-matrix models.

Contribution

It defines the (N,M)-th KdV hierarchy, explores its algebraic structures, reductions, and dispersionless limit, linking it to extended W algebras and multi-matrix models.

Findings

The hierarchy contains the higher KdV and multi-field KP as sub-systems.

It exhibits two compatible Poisson brackets generating extended W algebras.

Reductions lead to classical W_{N+M} algebras and relate to multi-matrix models.

Abstract

We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi--field representation of KP hierarchy as sub--systems and naturally appears in multi--matrix models. The N+2M-1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual $W_N$ algebra. We show that there exist M distinct reductions of the (N, M)--th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N, M) algebra is reduced to the W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.