Tau-functions and dressing transformations for zero-curvature affine integrable equations

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 g. 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 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 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 g. 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.

Construction of exact Riemannian instanton solutions

“…Such property is what makes the vertex operator representation to deserve the name of integral representation [17,18]. It also explain the truncation of the Hirota's expansion of the tau functions, since those are nothing more than special expectation values of V (µ) in the states of the vertex operator representation [21].…”

Vertex operator representation of the soliton tau functions in the An(1) Toda models by dressing transformations

“…The above expression together with (2.15a)-(2.15c) provides a hint of how to relate the dressing group approach to the group-algebraic methods, developed in [15,16]. This relation has been conjectured for general integrable hierarchies which admit a vacuum solution [13].…”

“…A deep relation between integrable hierarchies and the Kac-Moody (or affine) Lie algebras [10] has been clarified by Drinfeld and Sokolov [11]. In the last paper it was also explained the crucial role of Heisenberg subalgebras and the related to them gradations in constructing integrable evolution equations [12,13].…”

A n (1) Toda solitons and the dressing symmetry

“…In this paper we are not going to analyze this problem since it presents difficulties even for the dressing group elements which generate monosolitons from the vacuum in the sine-Gordon theory [15]. The dressing symmetry has been exploited recently [16] to treat a huge class of integrable hierarchies which admit a vacuum solution.…”

“…This problem has been already solved in [25] for the sine-Gordon equation which is the A (1) 1 Toda model. Expressions for the dressing group elements which create solitons from the vacuum have been conjectured in [16] for a large class of integrable hierarchies.…”