Tau functions for matrix hierarchies.

Abstract

The totality of all zero curvature equations with a rational dependence of connection matrices on a spectral parameter form a hierarchy, which means that all the corresponding vector fields commute. This is the so-called General Zakharov Shabat (GZS) hierarchy. We consider a subhierarchy of GZS with a given fixed set of poles. The "time variables" depend on three indices, one refers to a chosen pole, the other is a vector index taking values from 1 to n where n is a dimension of the matrices, and the third one corresponds to the order of the pole. In the case of a single pole, the subhierarchy is a generalization of the AKNS hierarchy with matrices of arbitrary dimension and a pole of arbitrary order.

The goal of the work is two-fold. First, we want to construct Grassmannian tau-functions for GZS. We present such a construction for its diagonal tau-functions. Second, we want to give an algebraic-geometrical construction of the Baker and tau-functions with a formula connecting them. We have considered the general case when the cross-poles equations are taken into account.