On WKB Theoretic Transformations for Painlevé Transcendents on Degenerate Stokes Segments

Abstract

The WKB theoretic transformation theorem established in [KT2] implies that the first Painlevé equation gives a normal form of Painlevé equations with a large parameter near a simple $P$-turning point. In this paper we extend this result and show that the second Painlevé equation $(P_{II})$ and the third Painlevé equation $(P_{II{I}^{\prime }(D_7)})$ of type $D_7$ give a normal form of Painlevé equations on a degenerate $P$-Stokes segments connecting two different simple $P$-turning points and on a degenerate $P$-Stokes segment of loop-type, respectively. That is, any 2-parameter formal solution of a Painlevé equation is reduced to a 2-parameter formal solution of $(P_{II})$ or $(P_{II{I}^{\prime }(D_7)})$ on these degenerate $P$-Stokes segments by our transformation.