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 ($J=\mathrm{I},\mathrm{II},\mathrm{IV};m=1;2;3;\dots$); the series is intended to clarify the structure of solutions of $(P_J{)}_m$ by the exact WKB analysis of the underlying overdetermined system $({\mathcal{D}\mathcal{SL}}_J{)}_m$ of linear differential 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 first kind, each instanton-type solution of $(P_J{)}_m$ 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 $({\mathcal{D}\mathcal{SL}}_J{)}_m$ to a solution of its canonical form $(\mathcal{D}\mathcal{C}an)$ (§5.3).