This is the third one of a series of articles on the exact WKB analysis of higher order Painlevé equations with a large parameter (J = I, II, IV; m = 1; 2; 3;…); the series is intended to clarify the structure of solutions of 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 . 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).
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Takahiro Kawai, Yoshitsugu Takei, WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton-type solutions of (J= I, 34, II-2 or IV) near a simple -turning point of the first kind. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 153–219DOI 10.2977/PRIMS/34