Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915226419462144 |
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| author | Alameddine, Mohamad Marchal, Olivier Nikolaev, Nikita Orantin, Nicolas |
| author_facet | Alameddine, Mohamad Marchal, Olivier Nikolaev, Nikita Orantin, Nicolas |
| contents | We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation Alameddine, Mohamad Marchal, Olivier Nikolaev, Nikita Orantin, Nicolas Differential Geometry Mathematical Physics Algebraic Geometry Classical Analysis and ODEs 34A26 (primary), 34M40, 34M55, 34M04, 34M25 We prove that the formal $\hbar$-power series solution of the deformed Painlevé I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface $x^5 + y^5 + z^5 = 0$ modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays. |
| title | Resurgence of Tritronquées Solutions of the Deformed Painlevé I Equation |
| topic | Differential Geometry Mathematical Physics Algebraic Geometry Classical Analysis and ODEs 34A26 (primary), 34M40, 34M55, 34M04, 34M25 |
| url | https://arxiv.org/abs/2503.12290 |