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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| Дата: | 2017 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2017
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/148731 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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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irk-123456789-1487312019-02-19T01:23:47Z Rational Solutions of the Painlevé-II Equation Revisited Miller, P.D. Sheng, Y. 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. 2017 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/148731 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Miller, P.D. Sheng, Y. |
| spellingShingle |
Miller, P.D. Sheng, Y. Rational Solutions of the Painlevé-II Equation Revisited Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Miller, P.D. Sheng, Y. |
| author_sort |
Miller, P.D. |
| title |
Rational Solutions of the Painlevé-II Equation Revisited |
| title_short |
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_sort |
rational solutions of the painlevé-ii equation revisited |
| publisher |
Інститут математики НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/148731 |
| 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. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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2023-05-20T17:31:13Z |
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2023-05-20T17:31:13Z |
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1796153474472214528 |