Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910793518284800 |
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| author | Long, Wen-Gao Jiang, Yun-Jiang Li, Yu-Tian |
| author_facet | Long, Wen-Gao Jiang, Yun-Jiang Li, Yu-Tian |
| contents | We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlevé transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex WKB method, also known as the method of uniform asymptotics, to approximate the second-order ODEs derived from PI's Lax pair with higher-order accuracy. As an application, we provide a rigorous proof of the full asymptotic expansion of the nonlinear eigenvalues proposed numerically by Bender, Komijani, and Wang. Additionally, we present the full asymptotic expansion for the pole parameters $(p_{n}, H_{n})$ corresponding to the $n$-th pole of the real tritronquée solution of the PI equation as $n \to +\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems Long, Wen-Gao Jiang, Yun-Jiang Li, Yu-Tian Exactly Solvable and Integrable Systems Classical Analysis and ODEs Complex Variables We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlevé transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex WKB method, also known as the method of uniform asymptotics, to approximate the second-order ODEs derived from PI's Lax pair with higher-order accuracy. As an application, we provide a rigorous proof of the full asymptotic expansion of the nonlinear eigenvalues proposed numerically by Bender, Komijani, and Wang. Additionally, we present the full asymptotic expansion for the pole parameters $(p_{n}, H_{n})$ corresponding to the $n$-th pole of the real tritronquée solution of the PI equation as $n \to +\infty$. |
| title | Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems |
| topic | Exactly Solvable and Integrable Systems Classical Analysis and ODEs Complex Variables |
| url | https://arxiv.org/abs/2405.19115 |