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 th...
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| Format: | Article |
| Language: | English |
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National Academy of Science of Ukraine
2007-03-01
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| Series: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Online Access: | http://www.emis.de/journals/SIGMA/2007/042/ |
| _version_ | 1828465113041993728 |
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| author | Kanehisa Takasaki |
| author_facet | Kanehisa Takasaki |
| author_sort | Kanehisa Takasaki |
| collection | DOAJ |
| description | 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). |
| first_indexed | 2024-12-11T03:26:38Z |
| format | Article |
| id | doaj.art-4bf234aa72ff4bd1a8121f71fe3560cf |
| institution | Directory Open Access Journal |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2024-12-11T03:26:38Z |
| publishDate | 2007-03-01 |
| publisher | National Academy of Science of Ukraine |
| record_format | Article |
| series | Symmetry, Integrability and Geometry: Methods and Applications |
| spelling | doaj.art-4bf234aa72ff4bd1a8121f71fe3560cf2022-12-22T01:22:29ZengNational Academy of Science of UkraineSymmetry, Integrability and Geometry: Methods and Applications1815-06592007-03-013042Hamiltonian Structure of PI HierarchyKanehisa TakasakiThe 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).http://www.emis.de/journals/SIGMA/2007/042/Painlevé equationsKdV hierarchyisomonodromic deformationsHamiltonian structureDarboux coordinates |
| spellingShingle | Kanehisa Takasaki Hamiltonian Structure of PI Hierarchy Symmetry, Integrability and Geometry: Methods and Applications Painlevé equations KdV hierarchy isomonodromic deformations Hamiltonian structure Darboux coordinates |
| title | Hamiltonian Structure of PI Hierarchy |
| title_full | Hamiltonian Structure of PI Hierarchy |
| title_fullStr | Hamiltonian Structure of PI Hierarchy |
| title_full_unstemmed | Hamiltonian Structure of PI Hierarchy |
| title_short | Hamiltonian Structure of PI Hierarchy |
| title_sort | hamiltonian structure of pi hierarchy |
| topic | Painlevé equations KdV hierarchy isomonodromic deformations Hamiltonian structure Darboux coordinates |
| url | http://www.emis.de/journals/SIGMA/2007/042/ |
| work_keys_str_mv | AT kanehisatakasaki hamiltonianstructureofpihierarchy |