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

### Takahiro Kawai

Kyoto University, Japan### Yoshitsugu Takei

Kyoto University, Japan

## 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 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 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).

## Cite this article

Takahiro Kawai, Yoshitsugu Takei, WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton-type solutions of $(P_J)_m$ (J= I, 34, II-2 or IV) near a simple $P$-turning point of the first kind. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 153–219

DOI 10.2977/PRIMS/34