Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 095, 24 pages      arXiv:1712.09033      https://doi.org/10.3842/SIGMA.2017.095

The Chazy XII Equation and Schwarz Triangle Functions

Oksana Bihun and Sarbarish Chakravarty
Department of Mathematics, University of Colorado, Colorado Springs, CO 80918, USA

Received June 21, 2017, in final form December 12, 2017; Published online December 25, 2017

Abstract
Dubrovin [Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348] showed that the Chazy XII equation $y'''- 2yy''+3y'^2 = K(6y'-y^2)^2$, $K \in \mathbb{C}$, 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 $y=a_1w_1+a_2w_2+a_3w_3$ where $w_i$ solve the generalized Darboux-Halphen system. This relationship holds only for certain values of the coefficients $(a_1,a_2,a_3)$ and the Darboux-Halphen parameters $(\alpha, \beta, \gamma)$, which are enumerated in Table 2. Consequently, the Chazy XII solution $y(z)$ is parametrized by a particular class of Schwarz triangle functions $S(\alpha, \beta, \gamma; z)$ which are used to represent the solutions $w_i$ of the Darboux-Halphen system. The paper only considers the case where $\alpha+\beta+\gamma$<$1$. 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 $(\hat{P}, \hat{Q},\hat{R})$.

Key words: Chazy; Darboux-Halphen; Schwarz triangle functions; hypergeometric.

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