KP Trigonometric Solitons and an Adelic Flag Manifold
We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wil...
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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/123456789/147795 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | KP Trigonometric Solitons and an Adelic Flag Manifold / L. Haine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862732352679051264 |
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| author | Haine, L. |
| author_facet | Haine, L. |
| citation_txt | KP Trigonometric Solitons and an Adelic Flag Manifold / L. Haine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165].
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| first_indexed | 2025-12-07T19:31:28Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147795 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T19:31:28Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Haine, L. 2019-02-16T08:22:43Z 2019-02-16T08:22:43Z 2007 KP Trigonometric Solitons and an Adelic Flag Manifold / L. Haine // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 21 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 35Q53; 37K10 https://nasplib.isofts.kiev.ua/handle/123456789/147795 We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165]. This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. I wish to thank S.N.M. Ruijsenaars for his comments about [6] during the ‘International Workshop on Special Functions, Orthogonal Polynomials, Quantum Groups and Related Topics’ dedicated to Dick Askey 70th birthday (Bexbach, October 2003), which hinted at some of the results presented here, as well as for sending [17]. I also thank two anonymous referees for stimulating suggestions, which led to improvement of the final form of the paper. Partial support from the European Science Foundation Programme MISGAM, the Marie Curie RTN ENIGMA and a Grant of the Belgian National Science Foundation (FNRS) are also gratefully acknowledged. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications KP Trigonometric Solitons and an Adelic Flag Manifold Article published earlier |
| spellingShingle | KP Trigonometric Solitons and an Adelic Flag Manifold Haine, L. |
| title | KP Trigonometric Solitons and an Adelic Flag Manifold |
| title_full | KP Trigonometric Solitons and an Adelic Flag Manifold |
| title_fullStr | KP Trigonometric Solitons and an Adelic Flag Manifold |
| title_full_unstemmed | KP Trigonometric Solitons and an Adelic Flag Manifold |
| title_short | KP Trigonometric Solitons and an Adelic Flag Manifold |
| title_sort | kp trigonometric solitons and an adelic flag manifold |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147795 |
| work_keys_str_mv | AT hainel kptrigonometricsolitonsandanadelicflagmanifold |