R-matrix construction of electromagnetic models for the Painlevé transcendents

Abstract

The Painlevé transcendents PI–PV 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 sl̃R*(2) of the loop algebra sl̃R(2). The Hamiltonians are obtained by composing elements of the Poisson commuting ring of spectral invariant functions on sl̃R*(2) with a time-dependent family of Poisson maps whose images are four-dimensional rational coadjoint orbits in sl̃R*(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. © 1995 American Institute of Physics.