Singular-perturbative Reduction to BIrkhoff Normal Form and Instanton-Type Formal Solutions of Hamiltonian Systems

Abstract

As is illustrated by the computation of monodromy groups of second-order Fuchsian equations (cf. [AKT1]), the exact WKB analysis provides us with a powerful tool for studying global behavior of solutions of linear ordinary differential equations. To generalize such an analysis to nonlinear equations, T. Kawai (RIMS, Kyoto Univ.), T. Aoki (Kinki Univ.) and the author have developed the WKB theory for Painleve equations with a large parameter in our series of articles ([KT1], [AKT2], [KT2]). (See [Tl], [T2] also.) In our treatment 2-parameter formal solutions called instanton-type solutions, which were constructed through the multiple-scale analysis in [AKT2], are playing a central role. Although we have succeeded in analyzing their local structure near simple turning points in [KT2], some of their important properties such as the behavior near fixed singular points have not been clarified yet. In this paper, to investigate their behavior near fixed regular-type singular points, we propose a new construction of 2-parameter formal solutions of Painleve equations with a large parameter. The new construction of formal solutions we propose here is based on Takano's work [Tkal] (see [Tka2] also), where he constructed a 2-parameter family of analytic solutions at each regular-type singular point of (ordinary) Painleve equations. He made use of the well-known fact that Painleve equations can be written in the form of Hamiltonian systems (which we call Painleve Hamiltonian systems here) and established some reduction theorem for