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

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.


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)
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