Multiple-scale analysis for Painlevé transcendents with a large parameter

Abstract

1. Introduction. The purpose of this note is to give a survey of a part of the article[AKT3] which is concerned with the exact WKB analysis of Painlev´e transcendents witha large parameter. The exact WKB analysis is an analysis based on the systematic useof WKB solutions and Borel resummed WKB solutions of differential equations. A WKBsolution is a kind of formal solution that is expanded in the (negative) power series ofa large parameter. Such a series is divergent in general, but is easily constructed. Bytaking Borel resummation of a WKB solution, we get a holomorphic solution to theoriginal equation. The correspondence between WKB solutions to holomorphic solutionsobtained by Borel resummation is, however, not so simple (connection problems). Ifone knows the correspondence completely, then one can obtain large amount of globalinformation about the solutions. In fact, we know the correspondence, at least generically,in the case of second order linear ordinary differential equations of Fuchsian type (witha large parameter) and we can calculate the monodromy groups of the equations (cf.[AKT2]). In [AKT3], we investigate the Painlev´e equations from such a point of view (cf.[KT] also). Thus we are interested in(i) constructing formal solutions of Painlev´e equations,(ii) solving the connection problems for these formal solutions.In this note, we focus on the former problem and we give an outline of the constructionof formal solutions of the Painlev´e equations.2. Formal solutions of the first Painlev´e equation.2.1 Formal solution without free parameter.Let us consider the first Painlev´e equation with a large parameter η: