Rational Solutions of the Painlevé-II Equation Revisited
The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert repr...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2017
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/148731 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Rational Solutions of the Painlevé-II Equation Revisited / P.D. Miller, Y. Sheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 39 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862547559272153088 |
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| author | Miller, P.D. Sheng, Y. |
| author_facet | Miller, P.D. Sheng, Y. |
| citation_txt | Rational Solutions of the Painlevé-II Equation Revisited / P.D. Miller, Y. Sheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 39 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.
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| first_indexed | 2025-11-25T16:03:57Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148731 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-25T16:03:57Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | Miller, P.D. Sheng, Y. 2019-02-18T18:15:38Z 2019-02-18T18:15:38Z 2017 Rational Solutions of the Painlevé-II Equation Revisited / P.D. Miller, Y. Sheng // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E17; 34M55; 34M56; 35Q15; 37K15; 37K35; 37K40 DOI:10.3842/SIGMA.2017.065 https://nasplib.isofts.kiev.ua/handle/123456789/148731 The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlevé-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlevé-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlevé-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method. This paper is a contribution to the Special Issue on Symmetries and Integrability of Dif ference Equations.
 The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html.
 P.D. Miller was supported during the preparation of this paper by the National Science Foundation under grant DMS-1513054. The authors are grateful to Thomas Bothner for many useful
 discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rational Solutions of the Painlevé-II Equation Revisited Article published earlier |
| spellingShingle | Rational Solutions of the Painlevé-II Equation Revisited Miller, P.D. Sheng, Y. |
| title | Rational Solutions of the Painlevé-II Equation Revisited |
| title_full | Rational Solutions of the Painlevé-II Equation Revisited |
| title_fullStr | Rational Solutions of the Painlevé-II Equation Revisited |
| title_full_unstemmed | Rational Solutions of the Painlevé-II Equation Revisited |
| title_short | Rational Solutions of the Painlevé-II Equation Revisited |
| title_sort | rational solutions of the painlevé-ii equation revisited |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148731 |
| work_keys_str_mv | AT millerpd rationalsolutionsofthepainleveiiequationrevisited AT shengy rationalsolutionsofthepainleveiiequationrevisited |