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 -turning point. In this paper we extend this result and show that the second Painlevé equation and the third Painlevé equation of type give a normal form of Painlevé equations on a degenerate -Stokes segments connecting two different simple -turning points and on a degenerate -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 or on these degenerate -Stokes segments by our transformation.
Cite this article
Kohei Iwaki, On WKB Theoretic Transformations for Painlevé Transcendents on Degenerate Stokes Segments. Publ. Res. Inst. Math. Sci. 51 (2015), no. 1 pp. 1–57