The Chazy XII Equation and Schwarz Triangle Functions

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2017
• Main Authors: Bihun, O., Chakravarty, S.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2017
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/149278
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: The Chazy XII Equation and Schwarz Triangle Functions / O. Bihun, S. Chakravarty // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ.

Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation y′′′−2yy′′+3y′²=K(6y′−y²)², K∈C, is equivalent to a projective-invariant equation for an affine connection on a one-dimensional complex manifold with projective structure. By exploiting this geometric connection it is shown that the Chazy XII solution, for certain values of K, can be expressed as y=a₁w₁+a₂w₂+a₃w₃ where wi solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients (a1,a2,a3) and the Darboux-Halphen parameters (α,β,γ), which are enumerated in Table 2. Consequently, the Chazy XII solution y(z) is parametrized by a particular class of Schwarz triangle functions S(α,β,γ;z) which are used to represent the solutions wi of the Darboux-Halphen system. The paper only considers the case where α+β+γ<1. The associated triangle functions are related among themselves via rational maps that are derived from the classical algebraic transformations of hypergeometric functions. The Chazy XII equation is also shown to be equivalent to a Ramanujan-type differential system for a triple (P^,Q^,R^).