WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton-type solutions of $(P_J)_m\ (J= \mathrm{I}, 34, \mathrm{II}-2$ or $\mathrm{IV})$ near a simple $P$-turning point of the first kind

Abstract

This is the third one of a series of articles on the exact WKB analysis of higher order Painlevé equations $(P\_J)\_m$ with a large parameter (J = I, II, IV; m = 1; 2; 3;…); the series is intended to clarify the structure of solutions of $(P\_J)\_m$ by the exact WKB analysis of the underlying overdetermined system (DSLJ)m of linear diff erential equations, and the target of this paper is instanton-type solutions of $(P\_J)\_m$. In essence, the main result (Theorem 5.1.1) asserts that, near a simple P-turning point of the rst kind, each instanton-type solution of (PJ )m can be formally and locally transformed to an appropriate solution of (\_P\_I)1, the classical (i.e., the second order) Painlevé-I equation with a large parameter. The transformation is attained by constructing a WKB-theoretic transformation that brings a solution of (DSLJ)m to a solution of its canonical form (DCan) (§5.3).