Asymptotics of solutions of the degenerate third Painlev\'e equation in the neighbourhood of the regular singular point: the isomonodromy deformation approach

A. V. Kitaev ; A. Vartanian

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 169-211

Résumé

This paper contains several technical refinements of our previously obtained results on the monodromy parametrisation of small-$\tau$ asymptotics of solutions $u(\tau)$ of the degenerate third Painlevé equation, $$ u^{\prime \prime}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau} \left(-8 \varepsilon (u(\tau))^{2} + 2ab \right) + \frac{b^{2}}{u(\tau)}, $$ where $\varepsilon = \pm 1$, $\varepsilon b > 0$, $a \in \mathbb{C},$ and of its associated mole function, $\varphi(\tau)$, which satisfies $\varphi^{\prime}(\tau) = \tfrac{2a}{\tau} + \tfrac{b}{u(\tau)}$. We also describe three families of three-real-parameter solutions $u(\tau)$ which have infinite sequences of zeros converging to the origin of the complex $\tau$-plane. Furthemore, for $a=0$, a numerical visualisation of the formulae connecting the asymptotics as $\tau\to0$ and $\tau\to+\infty$ of solutions $u(\tau)$ and $\varphi(\tau)$ having logarithmic behaviour as $\tau\to0$ is given.