Painlevé Equations

The special functions play a fundamental role in mathematical physics; some of them are solutions to linear differential equations [1]. In 1905, Paul Painlevé studied the nonlinear ordinary differential equations with specific mathematical properties. He obtained six of these equations, which can be considered as nonlinear analogues of the classical special functions. Lately, there has been an interest in these equations as they appear in several physical applications. Painlevé equations can be solved using algebraic techniques developed in quantum theory[2].