The Painleve functions are transcendental in general, but the Painleve equations have special solutions which are not transcendental. In this talk (2 hours), we give a definition of `special' solutions of the Painleve equations. In Umemura's sense, special solutions are divided into two classes. One is algebraic solutions and the other is so-called Riccati-type solutions. In order to give a precise definition of special solutions, we review Okamoto's initial value spaces. And the Backlund transformation groups play an important role to classify complete list of special solutions.
In the first lecture we show complete list of special solutions for P1 to P5. In the second lecture, we study a representation of special solutions. Moreover we may research other class of special solutions beyond Umemura's classical solutions, such as Boutroux's solutions (P1), Ablowits-Segal's solutions (P2), symmetric solutions (P1,2,4) and Picard's solution (P6).
This lecture is a bridge between Umemura's lecture and Boalch's lecture. Algebraic solutions of P6 are shown in Boalch's talk.
- http://www.math.sci.osaka-u.ac.jp/~ohyama/frame/newton/index.html - My web site on PEM