Systems of PDEs Obtained from Factorization in Loop Groups

摘要

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 g and a parabolic subalgebra p, 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 total hierarchy equals the rank of g. The choice of functions, however, is shown to depend in a noncanonical way on p. 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.