$R$--Matrix Construction of Electromagnetic Models for the Painlevé Transcendents

Abstract

The Painlevé transcendents $P_{\rom{I}}$--$P_{\rom{V}}$ and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical $R$--matrix Poisson bracket structure on the dual space $\wt{\frak{sl}}_R^*(2)$ of the loop algebra $\wt{\frak{sl}}_R(2)$. The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on $\wt{\frak{sl}}_R^*(2)$ with a time--dependent family of Poisson maps whose images are $4$--dimensional rational coadjoint orbits in $\wt{\frak{sl}}_R^*(2)$. Each system may be interpreted as describing a particle moving on a surface of zero curvature in the presence of a time--varying electromagnetic field. The Painlevé equations follow from reduction of these systems by the Hamiltonian flow generated by a second commuting element in the ring of spectral invariants.

22 pgs, AMSTeX, preprint CRM-2889 (1994)

References in corpus (1)

• Dual Isomonodromic Deformations and Moment Maps to Loop Algebras

Cited by in corpus (6)

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• Hamiltonian structure of rational isomonodromic deformation systems
• Algebras of quantum monodromy data and decorated character varieties
• Belinskii-Zakharov Formulation for Bianchi models and Painleve III Equation