Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter

Abstract

Various properties of algebroid solutions of the degenerate third Painlevé equation, \begin{equation*} u^{\prime \prime}(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)},\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 $\tau \to 0$, where $c\in \mathbb{C}\setminus \{0\}$, together with its associated integral ${\int }_0^{\tau }(u(t){)}^{-1}dt$, are considered in detail, and their basic asymptotic behaviours are visualized.

arXiv comment74 pages, 44 figures