Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 042, 32 pages      nlin.SI/0610073      http://dx.doi.org/10.3842/SIGMA.2007.042
Contribution to the Vadim Kuznetsov Memorial Issue

Hamiltonian Structure of PI Hierarchy

Kanehisa Takasaki
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida, Sakyo, Kyoto 606-8501, Japan

Received November 01, 2006, in final form February 13, 2007; Published online March 09, 2007

Abstract
The string equation of type (2,2g+1) may be thought of as a higher order analogue of the first Painlevé equation that corresponds to the case of g = 1. For g > 1, this equation is accompanied with a finite set of commuting isomonodromic deformations, and they altogether form a hierarchy called the PI hierarchy. This hierarchy gives an isomonodromic analogue of the well known Mumford system. The Hamiltonian structure of the Lax equations can be formulated by the same Poisson structure as the Mumford system. A set of Darboux coordinates, which have been used for the Mumford system, can be introduced in this hierarchy as well. The equations of motion in these Darboux coordinates turn out to take a Hamiltonian form, but the Hamiltonians are different from the Hamiltonians of the Lax equations (except for the lowest one that corresponds to the string equation itself).

Key words: Painlevé equations; KdV hierarchy; isomonodromic deformations; Hamiltonian structure; Darboux coordinates.

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