On the turning point problem for instanton-type solutions of Painlevé equations
نویسندگان
چکیده
The turning point problems for instanton-type solutions of Painlevé equations with a large parameter are discussed. Generalizing the main result of [KT2] near a simple turning point, we report in this paper that Painlevé equations can be transformed to the second Painlevé equation and the most degenerate third Painlevé equation near a double turning point and near a simple pole, respectively. An outline of the proof based on the theory of isomonodromic deformations of associated linear differential equations is also explained. 1 Background and main results The purpose of this report is to discuss the turning point problem for instanton-type solutions of Painlevé equations from the viewpoint of exact WKB analysis. In our series of papers ([KT1],[AKT],[KT2]) we develop the exact WKB analysis of Painlevé equations (PJ) with a large parameter η (> 0): (PJ) dλ dt2 = GJ ( λ, dλ dt , t ) + ηFJ(λ, t). ∗Supported in part by JSPS Grants-in-Aid No. 20340028 and No. 21340029.
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