Toward the Exact WKB Analysis of the PII Hierarchy

We develop the exact WKB analysis of the P _II hierarchy introduced by Gordoa et al. 5 ] in this paper; our particular interest is in the relation between the Stokes geometry of a higher order Painlevé equation in the hierarchy and that of its underlying Lax pair. An important observation is that Stokes curves of the Painlevé equation may cross, reflecting the higher order character of the equation, and in a neighborhood of the crossing point an unexpected degeneracy of the Stokes geometry of the Lax pair may occur along a curved ray emanating from the crossing point.