Virasoro Action and Virasoro Constraints on Integrable Hierarchies of the $r$-Matrix Type

Abstract

For a large class of hierarchies of integrable equations admitting a classical $r-$matrix, we propose a construction for the Virasoro algebra actionon the Lax operators which commutes with the hierarchy flows. The construction relies on the existence of dressing transformations associated to the $r$-matrix and does not involve the notion of a tau function. The dressing-operator form of the Virasoro action gives the corresponding formulation of the Virasoro constraints on hierarchies of the $r-$matrix type. We apply the general construction to several examples which include KP, Toda and generalized KdV hierarchies, the latter both in scalar and the Drinfeld-Sokolov formalisms. We prove the consistency of Virasoro action on the scalar and matrix (Drinfeld-Sokolov) Lax operators, and make an observation on the difference in the form of string equations in the two formalisms.