TAU-FUNCTIONS AND DRESSING TRANSFORMATIONS FOR ZERO-CURVATURE AFFINE INTEGRABLE EQUATIONS

The solutions of a large class of hierarchies of zero-curvature equati ons that includes Toda- and KdV-type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras g. Their common feature is that they have some spec ial ''vacuum solutions'' corresponding to Lax operators lying in some Abelian (up to the central term) subalgebra of g; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solu tions are constructed in a uniform way. Then, the generalized tau-func tions for those hierarchies are defined as an alternative set of varia bles corresponding to certain matrix elements evaluated in the integra ble highest-weight representations of g. Such definition of tau-functi ons applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the Abelian and non-Abelian affine Toda theories are discussed in derail. (C) 1997 American Institute of Physics.