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

We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{{}^{\mathrm{\prime }\mathrm{\prime }}}(\tau )=\frac{(u^{\mathrm{\prime }}(\tau ){)}^2}{u(\tau )}-\frac{u^{\mathrm{\prime }}(\tau )}{\tau }+\frac{1}{\tau }(-8\epsilon (u(\tau ){)}^2+2ab)+\frac{b^2}{u(\tau )},\epsilon =\pm 1,\epsilon b>0,a\in \mathbb{C}\setminus i\mathbb{Z}$, 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.