An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlevé Equations

Abstract

The classical Painlevé 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 Painlevé equations. In this paper, we review this method and indicate how it may be extended to the discrete Painlevé equations.Contents 6 4. Plane Algebraic Curves Arising in Limits 7 5. The first Painlevé equation in the limit t → ∞ in initial value space 9 6. The q-discrete first Painlevé equation in the limit t → ∞ 12 7. Discussion 13 References 13