Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation
It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212115 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation. Robert J. Buckingham and Peter D. Miller. SIGMA 20 (2024), 008, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862656163115433984 |
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| author | Buckingham, Robert J. Miller, Peter D. |
| author_facet | Buckingham, Robert J. Miller, Peter D. |
| citation_txt | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation. Robert J. Buckingham and Peter D. Miller. SIGMA 20 (2024), 008, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D₇) equation, valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside, to a form of the Weierstraß equation.
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| first_indexed | 2026-03-15T20:34:30Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212115 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T20:34:30Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Buckingham, Robert J. Miller, Peter D. 2026-01-28T13:57:30Z 2024 Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation. Robert J. Buckingham and Peter D. Miller. SIGMA 20 (2024), 008, 27 pages 1815-0659 2020 Mathematics Subject Classification: 34E05; 34M55; 37K10 arXiv:2308.16051 https://nasplib.isofts.kiev.ua/handle/123456789/212115 https://doi.org/10.3842/SIGMA.2024.008 It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D₇) equation, valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside, to a form of the Weierstraß equation. R.J. Buckingham was supported by the National Science Foundation under Grant DMS-2108019. P.D. Miller was supported by the National Science Foundation under Grants DMS-1812625 and DMS-2204896. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation Article published earlier |
| spellingShingle | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation Buckingham, Robert J. Miller, Peter D. |
| title | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation |
| title_full | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation |
| title_fullStr | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation |
| title_full_unstemmed | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation |
| title_short | Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III (D₇) Equation |
| title_sort | differential equations for approximate solutions of painlevé equations: application to the algebraic solutions of the painlevé-iii (d₇) equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212115 |
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