Painlevé transcendents

Summary

Painlevé transcendents are solutions of the six Painlevé equations $\mathrm{P_I}$ to $\mathrm{P_{VI}}$, nonlinear second order differential equations with the Painlevé property. Consider equations $y'' = R(t, y, y')$ with $R$ rational in $y$ and $y'$ that have this property. Up to changes of variables, the six equations are the only ones among them that are not solvable by earlier known functions. Their general solutions are new transcendental functions. They play for nonlinear problems the role that classical special functions play for linear ones, for instance in random matrix theory and integrable systems.