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Umeta, Yoko

General formal solutions for a unified family of \(P_{\mathrm{J}}\)-hierarchies (J=I, II, IV, 34). (English) Zbl 1435.34092

J. Math. Soc. Japan 71, No. 3, 979-1003 (2019).
To discuss several hierarchies of higher order Painlevé equations with a large parameter in a unified manner, the author first introduces a unified family of Painlevé hierarchies in this paper. A key idea for the formulation of the unified family is the use of generating functions for Painlevé hierarchies under consideration. Then, the author also succeeds in constructing instanton-type formal solutions, that is, formal solutions with different exponential terms that contain sufficiently many free parameters of the unified family. She constructs instanton-type formal solutions of the unified family by applying the multiple-scale method and solving differential equations describing the so-called non-secularity conditions, which are the key step in constructing formal solutions from this approach.
Reviewer: Yoshitsugu Takei (Kyoto)

Cited in 1 Review
Cited in 1 Document

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)

Keywords:

exact WKB analysis; Painlevé hierarchy; instanton-type solutions
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\textit{Y. Umeta}, J. Math. Soc. Japan 71, No. 3, 979--1003 (2019; Zbl 1435.34092)
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