1998/01/08 by J. Dorfmeister, Josef F. Dorfmeister, H. Gradl +5
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #nlin.SI #solv-int
paper · pdf · doi:10.48550/arxiv.solv-int/9801009
1 figure
arxiv created 1998/01/08 · openalex publication_date 1998/01/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose a generalization of a Drinfeld-Sokolov scheme of attaching integrable systems of PDEs to affine Kac-Moody algebras. With every affine Kac-Moody algebra ≫ and a parabolic subalgebra \gp, we associate two hierarchies of PDEs. One, called positive, is a generalization of the KdV hierarchy, the other, called negative, generalizes the Toda hierarchy. We prove a coordinatization theorem, which establishes that the number of functions needed to express all PDEs of the the total hierarchy equals the rank of ≫. The choice of functions, however, is shown to depend in a noncanonical way on \gp. We employ a version of the Birkhoff decomposition and a ``2-loop'' formulation which allows us to incorporate geometrically meaningful solutions to those hierarchies. We illustrate our formalism for positive hierarchies with a generalization of the Boussinesq system and for the negative hierarchies with the stationary Bogoyavlenskii equation.