Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions
The third Painlevé equation in its generic form, often referred to as Painlevé-III(₆), is given by d²/d² = 1/(d/d)² − 1/ d/d + (α² + β)/ + 4³ − 4/, α, β ∈ ℂ. Starting from a generic initial solution ₀() corresponding to parameters α, β, denoted as the triple (₀(), α, β), we apply an explicit Bäcklun...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212104 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions. Ahmad Barhoumi, Oleg Lisovyy, Peter D. Miller and Andrei Prokhorov. SIGMA 20 (2024), 019, 77 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862618970401538048 |
|---|---|
| author | Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei |
| author_facet | Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei |
| citation_txt | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions. Ahmad Barhoumi, Oleg Lisovyy, Peter D. Miller and Andrei Prokhorov. SIGMA 20 (2024), 019, 77 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The third Painlevé equation in its generic form, often referred to as Painlevé-III(₆), is given by d²/d² = 1/(d/d)² − 1/ d/d + (α² + β)/ + 4³ − 4/, α, β ∈ ℂ. Starting from a generic initial solution ₀() corresponding to parameters α, β, denoted as the triple (₀(), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (ₙ(), α+4, β+4) indexed by ∈ ℕ. We study the large n behavior of the solutions (ₙ(), α + 4, β +4n) under the scaling = / in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution ₙ(/). Our main result is a proof that the limit of solutions ₙ(z/) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(₈), d²/d² = 1/(d/dz)² − 1/ d/d + (4²+4)/. A notable application of our result is to rational solutions of Painlevé-III(₆), which are constructed using the seed solution (1, 4,−4) where ∈ ℂ∖(ℤ + 1/2) and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both ₆ and ₈, at = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of = 0.
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| first_indexed | 2026-03-14T11:24:10Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212104 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T11:24:10Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei 2026-01-28T13:55:34Z 2024 Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions. Ahmad Barhoumi, Oleg Lisovyy, Peter D. Miller and Andrei Prokhorov. SIGMA 20 (2024), 019, 77 pages 1815-0659 2020 Mathematics Subject Classification: 34M55; 34E05; 34M50; 34M56; 33E17 arXiv:2307.11217 https://nasplib.isofts.kiev.ua/handle/123456789/212104 https://doi.org/10.3842/SIGMA.2024.019 The third Painlevé equation in its generic form, often referred to as Painlevé-III(₆), is given by d²/d² = 1/(d/d)² − 1/ d/d + (α² + β)/ + 4³ − 4/, α, β ∈ ℂ. Starting from a generic initial solution ₀() corresponding to parameters α, β, denoted as the triple (₀(), α, β), we apply an explicit Bäcklund transformation to generate a family of solutions (ₙ(), α+4, β+4) indexed by ∈ ℕ. We study the large n behavior of the solutions (ₙ(), α + 4, β +4n) under the scaling = / in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution ₙ(/). Our main result is a proof that the limit of solutions ₙ(z/) exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III(₈), d²/d² = 1/(d/dz)² − 1/ d/d + (4²+4)/. A notable application of our result is to rational solutions of Painlevé-III(₆), which are constructed using the seed solution (1, 4,−4) where ∈ ℂ∖(ℤ + 1/2) and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at = 0 when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both ₆ and ₈, at = 0. We also deduce the large n behavior of the Umemura polynomials in a neighborhood of = 0. We would like to thank Roozbeh Gharakhloo and Deniz Bilman for bringing to our attention the applications of our work to 2j − k determinants and the Suleimanov solutions, respectively. We would like to thank Marco Fasondini for providing us with his program that we used to numerically confirm our results and produce Figures 1 and 2. The work of Andrei Prokhorov was supported by NSF MSPRF grant DMS-2103354, NSF grant DMS-1928930, and RSF grant 22-11-00070. Part of the work was done while Prokhorov was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2021 semester. The NSF partially supported Ahmad Barhoumi under grant DMS-1812625. The NSF partially supported Peter Miller under grants DMS-1812625 and DMS-2204896. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions Article published earlier |
| spellingShingle | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei |
| title | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| title_full | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| title_fullStr | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| title_full_unstemmed | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| title_short | Painlevé-III Monodromy Maps Under the ₆ → ₈ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| title_sort | painlevé-iii monodromy maps under the ₆ → ₈ confluence and applications to the large-parameter asymptotics of rational solutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212104 |
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