On the Increasing Tritronquée Solutions of the Painlevé-II Equation
The increasing tritronquée solutions of the Painlevé-II equation with parameter α exhibit square-root asymptotics in the maximally large sector |arg(x)| < 2/3π and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| 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/123456789/209879 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Increasing Tritronquée Solutions of the Painlevé-II Equation / P.D. Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862738168011292672 |
|---|---|
| author | Miller, P.D. |
| author_facet | Miller, P.D. |
| citation_txt | On the Increasing Tritronquée Solutions of the Painlevé-II Equation / P.D. Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The increasing tritronquée solutions of the Painlevé-II equation with parameter α exhibit square-root asymptotics in the maximally large sector |arg(x)| < 2/3π and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of α. Here, these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex α, all such solutions are asymptotically pole-free along the bisecting ray of the complementary sector |arg(− x)| < 1/3π that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity, and the poles are dealt with in the principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlevé-II parameter α is of the form α = ±1/2 + ip, p ∈ ℝ \ {0}, one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis.
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| first_indexed | 2025-12-07T20:02:58Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209879 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:02:58Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Miller, P.D. 2025-11-28T09:40:37Z 2018 On the Increasing Tritronquée Solutions of the Painlevé-II Equation / P.D. Miller // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E17; 34M40; 34M55; 35Q15 arXiv: 1804.03173 https://nasplib.isofts.kiev.ua/handle/123456789/209879 https://doi.org/10.3842/SIGMA.2018.125 The increasing tritronquée solutions of the Painlevé-II equation with parameter α exhibit square-root asymptotics in the maximally large sector |arg(x)| < 2/3π and have recently appeared in applications where it is necessary to understand the behavior of these solutions for complex values of α. Here, these solutions are investigated from the point of view of a Riemann-Hilbert representation related to the Lax pair of Jimbo and Miwa, which naturally arises in the analysis of rogue waves of infinite order. We show that for generic complex α, all such solutions are asymptotically pole-free along the bisecting ray of the complementary sector |arg(− x)| < 1/3π that contains the poles far from the origin. This allows the definition of a total integral of the solution along the axis containing the bisecting ray, in which certain algebraic terms are subtracted at infinity, and the poles are dealt with in the principal-value sense. We compute the value of this integral for all such solutions. We also prove that if the Painlevé-II parameter α is of the form α = ±1/2 + ip, p ∈ ℝ \ {0}, one of the increasing tritronquée solutions has no poles or zeros whatsoever along the bisecting axis. The author’s work was supported by the National Science Foundation under grants DMS1513054 and DMS-1812625. The author thanks Thomas Bothner, Deniz Bilman, and Liming Ling for useful discussions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Increasing Tritronquée Solutions of the Painlevé-II Equation Article published earlier |
| spellingShingle | On the Increasing Tritronquée Solutions of the Painlevé-II Equation Miller, P.D. |
| title | On the Increasing Tritronquée Solutions of the Painlevé-II Equation |
| title_full | On the Increasing Tritronquée Solutions of the Painlevé-II Equation |
| title_fullStr | On the Increasing Tritronquée Solutions of the Painlevé-II Equation |
| title_full_unstemmed | On the Increasing Tritronquée Solutions of the Painlevé-II Equation |
| title_short | On the Increasing Tritronquée Solutions of the Painlevé-II Equation |
| title_sort | on the increasing tritronquée solutions of the painlevé-ii equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209879 |
| work_keys_str_mv | AT millerpd ontheincreasingtritronqueesolutionsofthepainleveiiequation |