An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlev\'e Equations

Abstract

The classical Painlev\'e equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamoto's geometric space of initial values for the first and second Painlev\'e equations. In this paper, we review this method and indicate how it may be extended to the discrete Painlev\'e equations.