dorsal/arxiv
View SchemaSystems of PDEs obtained from factorization in loop groups
| Authors | J. Dorfmeister, H. Gradl, J. Szmigielski |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9801009 |
| URL | https://arxiv.org/abs/solv-int/9801009 |
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 $\gg$ 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 $\gg$. 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.
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"abstract": "We propose a generalization of a Drinfeld-Sokolov scheme of attaching\nintegrable systems of PDEs to affine Kac-Moody algebras. With every affine\nKac-Moody algebra $\\gg$ and a parabolic subalgebra $\\gp$, we associate two\nhierarchies of PDEs. One, called positive, is a generalization of the KdV\nhierarchy, the other, called negative, generalizes the Toda hierarchy. We prove\na coordinatization theorem, which establishes that the number of functions\nneeded to express all PDEs of the the total hierarchy equals the rank of $\\gg$.\nThe choice of functions, however, is shown to depend in a noncanonical way on\n$\\gp$. We employ a version of the Birkhoff decomposition and a ``2-loop\u0027\u0027\nformulation which allows us to incorporate geometrically meaningful solutions\nto those hierarchies. We illustrate our formalism for positive hierarchies with\na generalization of the Boussinesq system and for the negative hierarchies with\nthe stationary Bogoyavlenskii equation.",
"arxiv_id": "solv-int/9801009",
"authors": [
"J. Dorfmeister",
"H. Gradl",
"J. Szmigielski"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Systems of PDEs obtained from factorization in loop groups",
"url": "https://arxiv.org/abs/solv-int/9801009"
},
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