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arXiv · 2503.11912

The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

A. V. Kitaev·A. Vartanian
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Abstract

We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlev\'e equation, $u^{''}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau}\left(-8 \varepsilon (u(\tau))^{2} + 2ab \right) + \frac{b^{2}}{u(\tau)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$, 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.

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A. V. Kitaev, A. Vartanian. 2025-03-14. The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point. https://arxiv.org/abs/2503.11912

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