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

Abstract

We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{{}^{\prime \prime }}(\tau )=\frac{(u^{\prime }(\tau ){)}^2}{u(\tau )}-\frac{u^{\prime }(\tau )}{\tau }+\frac{1}{\tau }\left(-8\epsilon (u(\tau ){)}^2+2ab\right)+\frac{b^2}{u(\tau )},\epsilon =\pm 1,\epsilon b>0,a\in \mathbb{C}\setminus i\mathbb{Z}$, in terms of the monodromy data of a $2\times 2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.

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