Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy
Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as ''the coupled KP hierarchy'' and ''the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called '&#...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149097 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 33 назв. — англ. |
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Takasaki, K. 2019-02-19T17:19:31Z 2019-02-19T17:19:31Z 2009 Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 33 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35Q58; 37K10 https://nasplib.isofts.kiev.ua/handle/123456789/149097 Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as ''the coupled KP hierarchy'' and ''the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called ''the Pfaff-Toda hierarchy''). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve. The author is grateful to Ralf Willox and Saburo Kakei for useful information and comments. This work is partly supported by Grant-in-Aid for Scientific Research No. 19540179 and No. 21540218 from the Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy |
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Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy Takasaki, K. |
| title_short |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy |
| title_full |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy |
| title_fullStr |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy |
| title_full_unstemmed |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy |
| title_sort |
auxiliary linear problem, difference fay identities and dispersionless limit of pfaff-toda hierarchy |
| author |
Takasaki, K. |
| author_facet |
Takasaki, K. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as ''the coupled KP hierarchy'' and ''the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called ''the Pfaff-Toda hierarchy''). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.
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| issn |
1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/149097 |
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| citation_txt |
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy / K. Takasaki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 33 назв. — англ. |
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2025-11-24T02:40:13Z |
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2025-11-24T02:40:13Z |
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