The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Kitaev, A. V. Vartanian, A. |
| author_facet | Kitaev, A. V. Vartanian, A. |
| contents | We give a classification for the small-$τ$ asymptotic behaviours of solutions to the degenerate third Painlevé equation,
$u^{''}(τ) = \frac{(u^{\prime}(τ))^{2}}{u(τ)} - \frac{u^{\prime}(τ)}τ + \frac{1}τ\left(-8 \varepsilon (u(τ))^{2} + 2ab \right) + \frac{b^{2}}{u(τ)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$,
in terms of the monodromy data of a $2\times2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point Kitaev, A. V. Vartanian, A. Classical Analysis and ODEs 33E17, 34M30, 34M35, 34M40, 34M55, 34M56 We give a classification for the small-$τ$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{''}(τ) = \frac{(u^{\prime}(τ))^{2}}{u(τ)} - \frac{u^{\prime}(τ)}τ + \frac{1}τ\left(-8 \varepsilon (u(τ))^{2} + 2ab \right) + \frac{b^{2}}{u(τ)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$, in terms of the monodromy data of a $2\times2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions. |
| title | The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point |
| topic | Classical Analysis and ODEs 33E17, 34M30, 34M35, 34M40, 34M55, 34M56 |
| url | https://arxiv.org/abs/2503.11912 |