*(English)*Zbl 0878.34006

The authors discuss Bäcklund transformations and solution hierarchies for the third Painlevé equation ${\text{P}}_{\text{III}}$:

where $\alpha $, $\beta $, $\gamma $ and $\delta $ are arbitrary constants.

A survey of the study of Painlevé equations is given in Section 1. The integration of the continuous ${\text{P}}_{\text{III}}$ with $\beta =\delta =0$ or $\alpha =\gamma =0$ and several other properties are reviewed in Section 2, and many scaling transformations are also shown therein. In Section 3, various Bäcklund type transformations for ${\text{P}}_{\text{III}}$ are described. Section 4 is devoted to the parameter sets for which exact solutions of ${\text{P}}_{\text{III}}$ exist. In Section 5, these exact solutions are categorized into three hierarchies: solutions rational in $x$; solutions can be expressed by Bessel functions; and solutions rational in ${x}^{1/3}$.

In Section 5, the following discrete analogy of ${\text{P}}_{\text{III}}$ (d-P${}_{\text{III}}$)

is considered. The rational solution of (2) with $\gamma \delta \ne 0$ and exact solutions of (2) with $\gamma $ and $\delta $ being zero are considered. A final conclusion is stated in Section 6.

This long paper is interesting because the third Painlevé equation has a large number of physically significant applications. A bibliography of 78 papers is included.