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Symmetry, Integrability and Geometry: Methods and Applications, 2018, Volume 14, 075, 20 pp.
DOI: https://doi.org/10.3842/SIGMA.2018.075
(Mi sigma1374)
 

This article is cited in 5 scientific papers (total in 5 papers)

On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations

Anton Dzhamaya, Tomoyuki Takenawab

a School of Mathematical Sciences, The University of Northern Colorado, Campus Box 122, 501 20th Street, Greeley, CO 80639, USA
b Faculty of Marine Technology, Tokyo University of Marine Science and Technology, 2-1-6 Etchujima, Koto-ku Tokyo, 135-8533, Japan
Full-text PDF (501 kB) Citations (5)
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Abstract: 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 moreover, 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.
Keywords: integrable systems; Painlevé equations; difference equations; isomonodromic transformations; birational transformations.
Funding agency Grant number
Japan Society for the Promotion of Science 17K05271
A.D.’s work was partly supported by the University of Northern Colorado 2015 Summer Support Initiative. T.T. was supported by the Japan Society for the Promotion of Science, Grand-inAid (C) (17K05271).
Received: April 30, 2018; in final form July 14, 2018; Published online July 21, 2018
Bibliographic databases:
Document Type: Article
MSC: 34M55; 34M56; 14E07
Language: English
Citation: Anton Dzhamay, Tomoyuki Takenawa, “On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations”, SIGMA, 14 (2018), 075, 20 pp.
Citation in format AMSBIB
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\paper On Some Applications of Sakai's Geometric Theory of Discrete Painlev\'e Equations
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Symmetry, Integrability and Geometry: Methods and Applications
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