Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study
A one-parameter family of trans-series asymptotics as τ → ±∞ and τ → ±i∞ for solutions of the degenerate Painlevé III equation (DP3E), ′′(τ) = (′(τ))²/(τ) − ′(τ)/τ + 1/τ(−8ε(u(τ))² + 2) + ²/(τ), where ε ∈ {±1}, ∈ ℂ, and ∈ ℝ∖{0}, are parametrised in terms of the monodromy data of an associated firs...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/214097 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study. Arthur Vartanian. SIGMA 21 (2025), 067, 135 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | A one-parameter family of trans-series asymptotics as τ → ±∞ and τ → ±i∞ for solutions of the degenerate Painlevé III equation (DP3E), ′′(τ) = (′(τ))²/(τ) − ′(τ)/τ + 1/τ(−8ε(u(τ))² + 2) + ²/(τ), where ε ∈ {±1}, ∈ ℂ, and ∈ ℝ∖{0}, are parametrised in terms of the monodromy data of an associated first-order 2 × 2 matrix linear ODE via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the σ-forms of the DP3E are also obtained. The actions of various Lie-point symmetries for the DP3E are derived.
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| ISSN: | 1815-0659 |