Backlund transformation of Painleve III( D-8) inverted perpendicular function

M. A. Bershtein, A. I. Shchechkin

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14 Citations (Scopus)


We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(D 8) τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu-Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III(D 8) from the special representations of the Virasoro algebra of the highest weight (n + 1/4)2.

Original languageEnglish
Article number115205
JournalJournal of Physics A: Mathematical and Theoretical
Issue number11
Publication statusPublished - 20 Feb 2017


  • bilinear equations
  • Bäcklund transformations
  • Painlevé equations
  • Virasoro algebra


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