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Instanton-type solutions for the second and the fourth Painlevé hierarchies with a large parameter. (English) Zbl 1338.34168

Instanton-type formal solutions of higher order Painlevé equations were first constructed for the first Painlevé hierarchy by the reviewer [in: Algebraic analysis of differential equations. From microlocal analysis to exponential asymptotics. Festschrift in honor of Takahiro Kawai. Containing papers presented at the conference on algebraic analysis of differential equations – from microlocal analysis to exponential asymptotics, Kyoto, Japan, July 7–14, 2005. Tokyo: Springer. 307–319 (2007; Zbl 1184.34086)] via the Birkhoff normal form of Hamiltonian systems. After this work, T. Aoki, N. Honda and the author [Adv. Math. 235, 496–524 (2013; Zbl 1268.34185)] have computed their concrete forms by applying the multiple-scale analysis and introducing an idea of using their generating functions. In this paper, making use of the same method and idea as the above joint work with Aoki and Honda [loc. cit.], the author has succeeded in constructing instanton-type formal solutions explicitly for the second and fourth Painlevé hierarchies.

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
34E13 Multiple scale methods for ordinary differential equations
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References:

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