Abstract
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 \(\mathfrak{g} \) and a parabolic subalgebra \(\) , 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\(\mathfrak{g} \) . The choice of functions, however, is shown to depend in a noncanonical way on \(\). 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.
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Dorfmeister, J., Gradl, H. & Szmigielski, J. Systems of PDEs Obtained from Factorization in Loop Groups. Acta Applicandae Mathematicae 53, 1–58 (1998). https://doi.org/10.1023/A:1005947719355
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DOI: https://doi.org/10.1023/A:1005947719355