Discrete versions of the Painlevé equations

Abstract

We present discrete forms of the Painlev\'e transcendental equations ${\mathit{P}}_{\mathrm{III}}$,${\mathit{P}}_{\mathrm{IV}}$, and ${\mathit{P}}_{\mathrm{V}}$ that complement the list of the already known ${\mathit{P}}_{\mathrm{I}}$ and ${\mathit{P}}_{\mathrm{II}}$. These, most likely integrable, nonautonomous mappings go over to the usual Painlev\'e equations in the continuous limit, while in the autonomous limit we recover discrete system that belong to the integrable family of Quispel et al. Finally, we show that the discrete Painlev\'e mappings satisfy the same reduction relations as the continuous Painlev\'e transcendents, namely, ${\mathit{P}}_{\mathrm{V}}$\ensuremath{\rightarrow}{${\mathit{P}}_{\mathrm{III}}$, ${\mathit{P}}_{\mathrm{IV}}$}\ensuremath{\rightarrow}${\mathit{P}}_{\mathrm{II}}$\ensuremath{\rightarrow}${\mathit{P}}_{\mathrm{I}}$.