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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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2007 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2007
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147820 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Hamiltonian Structure of PI Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 41 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862644565561835520 |
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| author | Takasaki, K. |
| author_facet | Takasaki, K. |
| citation_txt | Hamiltonian Structure of PI Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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).
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| first_indexed | 2025-12-01T10:10:13Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147820 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T10:10:13Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Takasaki, K. 2019-02-16T08:51:28Z 2019-02-16T08:51:28Z 2007 Hamiltonian Structure of PI Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 41 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 34M55; 35Q53; 37K20 https://nasplib.isofts.kiev.ua/handle/123456789/147820 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). This paper is a contribution to the Vadim Kuznetsov Memorial Issue ‘Integrable Systems and Related Topics’. I would like to thank G. Falqui, E. Inoue and M. Mazzocco for valuable comments and discussions. This research was partially supported by Grant-in-Aid for Scientific Research No. 16340040 from the Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hamiltonian Structure of PI Hierarchy Article published earlier |
| spellingShingle | Hamiltonian Structure of PI Hierarchy Takasaki, K. |
| 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 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147820 |
| work_keys_str_mv | AT takasakik hamiltonianstructureofpihierarchy |