Parametric Stokes phenomenon for the second Painlevé equation with a large parameter

2011/06/03 by Kohei Iwaki, Iwaki, Kohei
Mathematics · #34M40 #34M55 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:34M40 #msc:34M55

paper · pdf · doi:10.48550/arxiv.1106.0612

49 pages, 35 figures

arxiv created 2011/06/20 · arxiv updated 2015/03/19

Abstract

The second Painlevé equation with a large parameter (P_II) is analyzed by using the exact WKB analysis. The purpose of this study is to investigate the problem of the degeneration of P-Stokes geometry of (P_II), which relates to a kind of Stokes phenomena for asymptotic (formal) solutions of (P_II). We call this Stokes phenomenon a "parametric Stokes phenomenon". We formulate the connection formula for this Stokes phenomenon, and confirm it in two ways: the first one is by computing the "Voros coefficient" of (P_II), and the second one is by using the isomonodromic deformation theory. Our main claim is that the connection formulas derived by these two completely different methods coincide.

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