Integrable Systems and WKB Analysis

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In this project, with the aid of Prof. T. Kawai (RIMS, Kyoto Univ.) and Mr. Y. Nishikawa (a graduate student of Kyoto Univ.), we have mainly studied generalization of exact WKB analysis to higher order Painleve equations that are obtained from integrable systems. First, for several hierarchies of higher order Painleve equations such as the (P_I) hierarchy obtained from the most degenerate Garnier system, the (P_<II>) hierarchy obtained by reduction of the KdV hierarchy, and the Noumi-Yamada system (i.e., (P_<IV>) and (P_V) hierarchy), we have found that turning points and Stokes curves ("Stokes geometry") of these nonlinear equations are closely related with those of the associated Lax pair. Secondly, it is discovered that Stokes curves of higher order Painleve equations do cross and a new Stokes curve emanates from a crossing point. A new Stokes curve can be understood as a Stokes curve emanating from a virtual turning point, which is also characterized in terms of the Stokes geometry of the associated Lax pair. Lastly, we have shown that a 0-parameter solution (i.e., an algebraically constructed formal solution without any free parameter) of any member of (P_I) and (P_<II>) hierarchies can be locally transformed to that of the traditional (P_I) equation near a simple turning point of the first kind. To examine if these results hold for more general nonlinear equations arising as compatibility condition of Lax pairs will be a main problem in future. In parallel with the above researches, we have also studied the following related subjects: (i) refinement of the exact steepest descent method, a method detecting new Stokes curves of linear equations, (ii) exact WKB analysis for systems of linear equations and its application to computations of transition probabilities, and (iii) exact WKB analysis for microdifferential equations of WKB type.