6 Hamiltonian structure of PI hierarchy Kanehisa Takasaki
نویسنده
چکیده
The string equation of type (2, 2g+1) may be thought of as a higher order analogue of the first Painlevé equation that correspond 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). The difference originates in the presence of extra terms in the isomonodromic Lax equations.
منابع مشابه
Hamiltonian Structure of PI Hierarchy
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. ...
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