On the Stokes geometry of higher order Painlevé equations

We show several basic properties concerning the relation between the Stokes geometry (i.e., configuration of Stokes curves and turning points) of a higher order Painlevé equation with a large parameter and the Stokes geometry of (one of) the underlying Lax pair. The higherorder Painlevé equation with a large parameter to be considered in this paper is one of the members of PJ -hierarchy with J =I,II-1 or II2, which are concretely given in Section 1. Since we deal with higher order equations, the Stokes curves may cross; some anomaly called the Nishikawa phenomenon may occur at the crossing point, and in this paper we analyze the mechanism why and how the Nishikawa phenomenon occurs. Several examples of Stokes geometry are given in Section 5 to visualize the core part of our results. 0 Introduction This paper is the first of a series of our papers on the exact WKB analysis of higher order Painlevé equations. For the sake of the clarity and the uniformity of the description we restrict our consideration in this paper to the PI, PII-1 and PII-2 hierarchies with a large parameter η, which are described explicitly in Section 1. Although these hierarchies are basically the same as those discussed by Shimomura ([S2]), Gordoa-Pickering ([GP]) and Gordoa-Joshi-Pickering ([GJP]), we need to appropriately introduce a large parameter η in their coefficients together with the underlying systems of linear differential equations (the so-called Lax pairs) so that we may develop the WKB analysis of the hierarchies in question. As is evident in the series of papers ([KT1], [AKT2], [KT2], [T1]; see [KT3] for their résumé), the relations between the Stokes geometry for (one of) the Lax pair and the appropriately defined Stokes geometry for the Painlevé equation play the key role in the WKB analysis of the traditional Painlevé equations, i.e., the second order differential equations first studied by Painlevé and Gambier. One of our main purposes of this paper is to show that the relations observed for the traditional Painlevé equations remain to hold for each member in the Painlevé hierarchies considered in this paper (Section 2). Another main purpose of this paper is to analyze why the novel and interesting phenomena numerically discovered by one of us (Y.N.) should occur in our context (Section 3). To analytically detect where the phenomena (the so-called Nishikawa phenomena) are observed, we introduce the notion of new Stokes curves in Section 4.