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Applied Mathematics Seminar
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Takahiro Kawai and Yoshitsugu Takei
RIMS, Kyoto University

On the Toulouse Project - how much we have understood the structure of solutions of higher order Painleve equations

Wednesday 10th September 14:05-14:55pm, Eastern Avenue Lecture Theatre.

The Toulouse Project is a collection of problems/proposals whose solutions should help us to understand the structure of solutions of higher order Painlev\'e equations $(P_J)_m$ ($J=I,II,IV$) introduced by P.R. Gordoa, N. Joshi and A. Pickering (Publ. RIMS, 37(2001), 327-347) and S. Shimomura (Ann. Scuola Norm. Sup. Pisa, 29(2000), 1-17). Its original form consists of seven steps, and we are in the middle stage of the fifth step, that is, we have clarified the structure of instanton-type solutions of $(P_J)_m$ near a turning point of the first kind, whose definition will be given in the talk.

In this talk we discuss WKB theoretic aspects of our theory, emphasizing the importance of WKB analysis of the underlying Lax pair, the simultaneous equations whose compatibility condition coincides with $(P_J)_m$. We also plan to discuss what points are peculiar to higher order Painlev\'e equations, i.e., not observed in the second order ones.