On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations
Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the cl...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Sprache: | English |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209775 |
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| Zitieren: | On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations / A. Dzhamay, T. Takenawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
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Dzhamay, A. Takenawa, T. 2025-11-26T11:35:17Z 2018 On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations / A. Dzhamay, T. Takenawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 34M56; 14E07 arXiv: 1804.10341 https://nasplib.isofts.kiev.ua/handle/123456789/209775 https://doi.org/10.3842/SIGMA.2018.075 Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system, and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type, and we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples. A.D.’s work was partly supported by the University of Northern Colorado's 2015 Summer Support Initiative. T.T. was supported by the Japan Society for the Promotion of Science, Grant-in-Aid (C) (17K05271). We thank N. Nakazono for explaining to us the techniques discussed in Remark 4.4. We are very grateful to A. Ramani, R. Willox, and the referees for their useful suggestions and corrections. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations |
| spellingShingle |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations Dzhamay, A. Takenawa, T. |
| title_short |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations |
| title_full |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations |
| title_fullStr |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations |
| title_full_unstemmed |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations |
| title_sort |
on some applications of sakai's geometric theory of discrete painlevé equations |
| author |
Dzhamay, A. Takenawa, T. |
| author_facet |
Dzhamay, A. Takenawa, T. |
| publishDate |
2018 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system, and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type, and we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/209775 |
| citation_txt |
On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations / A. Dzhamay, T. Takenawa // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 24 назв. — англ. |
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AT dzhamaya onsomeapplicationsofsakaisgeometrictheoryofdiscretepainleveequations AT takenawat onsomeapplicationsofsakaisgeometrictheoryofdiscretepainleveequations |
| first_indexed |
2025-12-07T16:57:32Z |
| last_indexed |
2025-12-07T16:57:32Z |
| _version_ |
1850886105439666176 |