Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlevé Equation. I

Abstract

The degenerate third Painlevé equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}τ + \frac{1}τ(-8 εu^{2} + 2ab) + \frac{b^{2}}{u}$, where $ε,b \in \mathbb{R}$, and $a \in \mathbb{C}$, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as $τ\to \pm 0$ and $\pm i0$ solution and general regular as $τ\to \pm \infty$ and $\pm i \infty$ solution are obtained.