Trans-Series Asymptotics of Solutions to the Degenerate Painlevé III Equation: A Case Study

A one-parameter family of trans-series asymptotics as $\tau \to \pm \infty$ and $\tau \to \pm {\rm i} \infty$ for solutions of the degenerate Painlevé III equation (DP3E), $$ u^{\prime \prime}(\tau) = \frac{(u^{\prime} (\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau}\bigl(-8 \varepsilon (u(\tau))^{2} + 2ab\bigr) + \frac{b^{2}}{u(\tau)},$$ where $\varepsilon \in \lbrace \pm 1 \rbrace$, $a \in \mathbb{C}$, and $b \in \mathbb{R} \setminus \lbrace 0 \rbrace$, are parametrised in terms of the monodromy data of an associated first-order $2 \times 2$ matrix linear ODE via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the $\sigma$-forms of the DP3E are also obtained. The actions of various Lie-point symmetries for the DP3E are derived.