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Volume 47, Issue 1, 2011, pp. 153–219
DOI: 10.2977/PRIMS/34

Published online: 2011-03-24

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[1] and Yoshitsugu Takei[2]

(1) Kyoto University, Japan
(2) Kyoto University, Japan

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 (PI)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).

Keywords: higher order Painlevé equations, Painlevé hierarchy, exact WKB analysis, instanton-type solutions, P-turning points

Kawai Takahiro, Takei Yoshitsugu: 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), 153-219. doi: 10.2977/PRIMS/34