One-Parameter Meromorphic Solution of the Degenerate Third Painlevé Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin

Abstract

We prove that there exists a one-parameter meromorphic solution $u(\tau )$ vanishing at $\tau =0$ 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 formal monodromy parameter $a=\pm i/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(\tau )$ at $\tau =0$ and its asymptotic behaviour as $\tau \to +\infty$. These asymptotics are visualized for generic initial data.

28 pages, 8 figures