2013/11/25 by Nalini Joshi, Joshi, Nalini
Mathematics · Physics and Astronomy · #34M30 #34M55 #39A13 #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.1311.6194
openalex publication_date 2013/11/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.