Equations Equivalent to the Painlevé Equations

Abstract

The Painlevé equations are nonlinear second-order ordinary differential equations which are studied and applied for analyzing physical phenomena in many fields of Physics including statistical mechanics, plasma physics, nonlinear waves and quantum field theory. This research is devoted to the equivalence problems of the Painlevé equations under a general point transformation. The necessary and sufficient conditions that an equation of the form y" = F (x,y,y') is to be equivalent to the Painlevé equations are found.