The authors discuss Bäcklund transformations and solution hierarchies for the third Painlevé equation :
where , , and are arbitrary constants.
A survey of the study of Painlevé equations is given in Section 1. The integration of the continuous with or 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 are described. Section 4 is devoted to the parameter sets for which exact solutions of exist. In Section 5, these exact solutions are categorized into three hierarchies: solutions rational in ; solutions can be expressed by Bessel functions; and solutions rational in .
In Section 5, the following discrete analogy of (d-P)
is considered. The rational solution of (2) with and exact solutions of (2) with and 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.