General formal solutions for a unified family of \(P_{\mathrm{J}}\)-hierarchies (J=I, II, IV, 34) (Q2330995)

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.