Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane
In this paper, we obtain large z asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as n × n Wronskian determinants involving classical Airy funct...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209850 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane / A. Deaño // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 35 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862560210281824256 |
|---|---|
| author | Deaño, A. |
| author_facet | Deaño, A. |
| citation_txt | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane / A. Deaño // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 35 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we obtain large z asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as n × n Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as n-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane.
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| first_indexed | 2025-12-04T11:29:22Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209850 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-04T11:29:22Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Deaño, A. 2025-11-27T14:53:57Z 2018 Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane / A. Deaño // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 35 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34M55; 34E05; 33C10; 30E10 arXiv: 1804.00563 https://nasplib.isofts.kiev.ua/handle/123456789/209850 https://doi.org/10.3842/SIGMA.2018.107 In this paper, we obtain large z asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as n × n Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as n-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane. The author acknowledges financial support from the EPSRC grant "Painlevé equations: analytical properties and numerical computation", reference EP/P026532/1, and from the project MTM2015-65888-C4-2-P from the Spanish Ministry of Economy and Competitiveness. The author wishes to thank M. Fasondini, D. Huybrechs, A. Iserles, A.R. Its, A.B.J. Kuijlaars, A.F. Loureiro, C. Pech, and W. Van Assche for stimulating discussions on the topic and scope of this paper, as well as the organizers of the workshop "Painlevé Equations and Applications" held at the University of Michigan, August 25-29, 2017, for their hospitality. The comments, remarks, and corrections of the anonymous referees have led to an improved version of the paper, and they are greatly appreciated. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane Article published earlier |
| spellingShingle | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane Deaño, A. |
| title | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane |
| title_full | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane |
| title_fullStr | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane |
| title_full_unstemmed | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane |
| title_short | Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane |
| title_sort | large z asymptotics for special function solutions of painlevé ii in the complex plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209850 |
| work_keys_str_mv | AT deanoa largezasymptoticsforspecialfunctionsolutionsofpainleveiiinthecomplexplane |