The Discrete Painlevé I Hierarchy

Abstract

The discrete Painlevé I equation (dP${}_{\mathrm{I}}$) is an integrable difference equation which has the classical first Painlevé equation (P${}_{\mathrm{I}}$) as a continuum limit. dP${}_{\mathrm{I}}$ is believed to be integrable because it is the discrete isomonodromy condition for an associated (single-valued) linear problem. In this paper, we derive higher-order difference equations as isomonodromy conditions that are associated to the same linear deformation problem. These form a hierarchy that may be compared to hierarchies of integrable ordinary differential equations (ODEs). We strengthen this comparison by continuum limit calculations that lead to equations in the P${}_{\mathrm{I}}$ hierarchy. We propose that our difference equations are discrete versions of higher-order Painlevé equations.

9 pages in LaTeX. To appear in Proceedings of SIDEII, Kent, UK 1996, (eds) P.A.Clarkson and F.Nijhoff