JournalsprimsVol. 51 , No. 1pp. 1–57

On WKB Theoretic Transformations for Painlevé Transcendents on Degenerate Stokes Segments

  • Kohei Iwaki

    Kyoto University, Japan
On WKB Theoretic Transformations for Painlevé Transcendents on Degenerate Stokes Segments cover
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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 PP-turning point. In this paper we extend this result and show that the second Painlevé equation (PII)(P_{II}) and the third Painlevé equation (PIII(D7))(P_{III'(D_7)}) of type D7D_7 give a normal form of Painlevé equations on a degenerate PP-Stokes segments connecting two different simple PP-turning points and on a degenerate PP-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 (PII)(P_{II}) or (PIII(D7))(P_{III'(D_7)}) on these degenerate PP-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

DOI 10.4171/PRIMS/148