On a unified approach to transformations and elementary solutions of Painlevé equations

Abstract

An algorithmic method is developed for investigating the transformation properties of second‐order equations of Painlevé type. This method, which utilizes the singularity structure of these equations, yields explicit transformations which relate solutions of the Painlevé equations II–VI, with different parameters. These transformations easily generate rational and other elementary solutions of the equations. The relationship between Painlevé equations and certain new equations quadratic in the second derivative of Painlevé type is also discussed.