Instanton-type solutions for the second and the fourth Painlevé hierarchies with a large parameter (Q500080)

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, \textit{T. Aoki}, \textit{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.