Constrained KP Models as Integrable Matrix Hierarchies

TL;DR

This paper formulates the constrained KP hierarchy as an integrable matrix hierarchy based on affine sl(M+K+1), providing new algebraic results including explicit recursion operators and characterizations of isospectral flows.

Contribution

It introduces a novel algebraic formulation of the constrained KP hierarchy as an affine matrix integrable hierarchy, deriving universal formulas and explicit recursion operators.

Findings

Closed expression for the second bracket via Dirac reduction.

Explicit recursion operator for the affine hierarchy.

Characterization of isospectral flows based on semisimple elements.

Abstract

We formulate the constrained KP hierarchy (denoted by \cKP$_{K+1,M}$) as an affine ${\widehat {sl}} (M+K+1)$ matrix integrable hierarchy generalizing the Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld-Sokolov hierarchy, we are able to find several new universal results valid for the \cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac-Moody current algebra. An explicit example is given for the case ${\widehat {sl}} (M+K+1)$, for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple {\em non-regular} element $E$ of $sl (M+K+1)$ and the content of the center of the kernel of $E$.