Truncation-type methods and Bäcklund transformations for ordinary differential equations: The third and fifth Painlevé equations (Q5890312)

In the recent paper [Nonlinearity 12, No. 4, 955-968 (1999; Zbl 0945.34073)], the authors presented a truncation-type method of deriving Bäcklund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the moveable poles that the equation possesses.NEWLINENEWLINENEWLINEIn this paper, the authors apply this approach to the third and fifth Painlevé equations. For the third Painlevé equation, they are able to obtain all fundamental Bäcklund transformations for the case where the parameters satisfy \(\delta \neq 0\). For the fifth Painlevé equation, their approach yields, what appears to be all known, Bäcklund transformations.