Classification of Painlevé type equations by the Mei symmetries and their exact solutions

Abstract

There are fifty second-order ordinary differential equations of Painlevé type in the literature. This paper investigates the Mei symmetries corresponding to the Lagrangian of these equations. This classifies these equations by the admitted Mei symmetries, along with the Mei invariants and their respective gauge functions. The existence of Mei symmetries can be correlated with the autonomous and non-autonomous properties of ordinary differential equations of Painlevé type. Furthermore, using symmetries, exact solutions to certain Painlevé type equations are obtained.