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~$⋙$. 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~$⋙$; 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~$⋙$. 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.

27 pages, plain TeX

References in corpus (1)

• The Symmetric Space and Homogeneous sine-Gordon Theories

Cited by in corpus (38)

• A New Approach to Integrable Theories in any Dimension
• Integrable Deformations of Strings on Symmetric Spaces
• Pohlmeyer reduction revisited
• Semi-classical spectrum of the Homogeneous sine-Gordon theories
• On generating functions in the AKNS hierarchy
• A simple formula for the conserved charges of soliton theories
• The complex Sine-Gordon equation as a symmetry flow of the AKNS Hierarchy
• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• Confinement, solitons and the equivalence between the sine-Gordon and massive Thirring models
• Toda equations associated with loop groups of complex classical Lie groups
• The Bullough-Dodd model coupled to matter fields
• Negative Even Grade mKdV Hierarchy and its Soliton Solutions
• Solitons, kinks and extended hadron model based on the generalized sine-Gordon theory
• Confinement and soliton solutions in the SL(3) Toda model coupled to matter fields
• Soliton Solutions for the Super mKdV and sinh-Gordon Hierarchy
• Symmetries and exact solutions of the BPS Skyrme model
• Grassmanian and Bosonic Thirring Models with Jump Defects
• Dyonic Integrable Models
• Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations
• Nonautonomous mixed mKdV-sinh-Gordon hierarchy
• Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation
• A Class of Mixed Integrable Models
• New derivation of soliton solutions to the AKNS${}_2$ system via dressing transformation methods
• Construction of exact Riemannian instanton solutions
• Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy
• Vertex Operators and Solitons of Constrained KP Hierarchies
• Supersymmetry for integrable hierarchies on loop superalgebras
• Solitons with Isospin
• Vertex Operators and Soliton Solutions of Affine Toda Model with U(2) Symmetry
• $A_n^{(1)}$ Toda solitons and the dressing symmetry
• Gauging of Geometric Actions and Integrable Hierarchies of KP Type
• More non-Abelian loop Toda solitons
• Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy
• Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies
• Solving non-abelian loop Toda equations
• Vertex Operator Representation of the Soliton Tau Functions in the $A_n^{(1)}$ Toda Models by Dressing Transformations
• New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations
• New negative grade solitonic sector for supersymmetric KdV and mKdV hierarchies