# 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 $(DSL_{J})_{m}$ 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 $(DSL_{J})_{m}$ 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