Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations

TL;DR

This paper develops a unified approach using tau-functions and dressing transformations to analyze solutions of zero-curvature integrable hierarchies derived from affine Kac-Moody algebras, including Toda and KdV types.

Contribution

It introduces a general framework for constructing solutions and tau-functions for affine integrable hierarchies using dressing transformations and highest-weight representations.

Findings

Uniform construction of solutions via dressing transformations.

Definition of tau-functions as matrix elements in highest-weight representations.

Application to affine Toda, mKdV, and KdV hierarchies.

Abstract

The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$\ggg$. Their common feature is that they have some special ``vacuum solutions'' corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$\ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$\ggg$. Such definition of tau-functions 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 detail.