Algebroid Solutions of the Degenerate Third Painlev\'e Equation for Vanishing Formal Monodromy Parameter

Abstract

Various properties of algebroid solutions of the degenerate third Painlev\'e equation, \begin{equation*} u^{\prime \prime}(\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)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{equation*} for the monodromy parameter $a=0$ are studied. The paper contains connection results for asymptotics as $\tau\to+0$ and as $\tau\to+\infty$ for $a\in\mathbb{C}$. Using these results, the simplest algebroid solution with asymptotics $u(\tau)\to c\tau^{1/3}$ as $\tau\to0$, where $c\in\mathbb{C}\setminus\{0\}$, together with its associated integral $\int_0^\tau {(u(t))^{-1}\,d t}$, are considered in detail, and their basic asymptotic behaviours are visualized.