Painlevé equations

The special functions play a fundamental role in mathematical physics; some are solutions to linear differential equations. In 1905, Paul Painlevé studied the nonlinear ordinary differential equations with specific mathematical properties. He obtained six of these equations, which can be considered nonlinear analogues of the classical special functions. Lately, these equations have been intriguing as they appear in several physical applications. Painlevé equations can be solved using algebraic techniques developed in quantum theory [Bermudez2013].

References:

[Bermudez2013] D. Bermudez, Supersymmetric quantum mechanics and Painleve equations, PhD Thesis, Cinvestav (2013).