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SIGMA 3 (2007), 042, 32 pages nlin.SI/0610073
https://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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