Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II

TL;DR

This paper derives asymptotic formulas for solutions of the degenerate third Painleve' equation using the Isomonodromy Deformation Method, revealing detailed behaviors of solutions and their zeroes and poles along axes.

Contribution

It provides new connection formulae for the asymptotics of solutions of the degenerate third Painleve' equation, linking them to monodromy data.

Findings

Asymptotics of regular and singular solutions as t approaches infinity and imaginary infinity are obtained.

Families of solutions with infinite zeroes and poles along axes are characterized.

Asymptotic locations of zeroes and poles are explicitly described.

Abstract

The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t -> +/-\infty$ and $t -> +/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotically located along the real and imaginary axes are distinguished: asymptotics of these zeroes and poles are also obtained.