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

SIGMA 13 (2017), 095, 24 pages      arXiv:1712.09033

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

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.

pdf (523 kb)   tex (46 kb)


  1. Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation and Schwarzian triangle functions, Asian J. Math. 2 (1998), 619-624.
  2. Ablowitz M.J., Chakravarty S., Halburd R., The generalized Chazy equation from the self-duality equations, Stud. Appl. Math. 103 (1999), 75-88.
  3. Atiyah M., Hitchin N., The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.
  4. Bureau F.J., Integration of some nonlinear systems of ordinary differential equations, Ann. Mat. Pura Appl. 94 (1972), 345-359.
  5. Chakravarty S., Differential equations for triangle groups, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 179-204.
  6. Chakravarty S., Ablowitz M.J., Integrability, monodromy evolving deformations, and self-dual Bianchi IX systems, Phys. Rev. Lett. 76 (1996), 857-860.
  7. Chakravarty S., Ablowitz M.J., Parameterizations of the Chazy equation, Stud. Appl. Math. 124 (2010), 105-135, arXiv:0902.3468.
  8. Chakravarty S., Ablowitz M.J., Clarkson P.A., Reductions of self-dual Yang-Mills fields and classical systems, Phys. Rev. Lett. 65 (1990), 1085-1087.
  9. Chakravarty S., Ablowitz M.J., Takhtajan L.A., Self-dual Yang-Mills equation and new special functions in integrable systems, in Nonlinear Evolution Equations and Dynamical Systems (Baia Verde, 1991), World Sci. Publ., River Edge, NJ, 1992, 3-11.
  10. Chazy J., Sur les équations différentielles du troisième ordre et d'ordre supérieur dont l'intégrale générale a ses points critiques fixes, Acta Math. 34 (1911), 317-385.
  11. Clarkson P.A., Olver P.J., Symmetry and the Chazy equation, J. Differential Equations 124 (1996), 225-246.
  12. Cosgrove C.M., Chazy classes IX-XI of third-order differential equations, Stud. Appl. Math. 104 (2000), 171-228.
  13. Darboux G., Mémoire sur la théorie des coordonnées curvilignes, et des systèmes orthogonaux, Ann. Sci. École Norm. Sup. (2) 7 (1878), 101-150.
  14. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  15. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. 1, McGraw-Hill, 1953.
  16. Ferapontov E.V., Galv\ ao C.A.P., Mokhov O.I., Nutku Y., Bi-Hamiltonian structure of equations of associativity in $2$-d topological field theory, Comm. Math. Phys. 186 (1997), 649-669.
  17. Ford L.R., Automorphic functions, 2nd ed., Chelsea Publishing Co., New York, 1951.
  18. Goursat E., Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique, Ann. Sci. École Norm. Sup. (2) 10 (1881), 3-142.
  19. Halphen G., Sur une système d'équations différentielles,, C.R. Acad. Sci. Paris 92 (1881), 1101-1103.
  20. Halphen G., Sur certains système d'équations différentielles, C.R. Acad. Sci. Paris 92 (1881), 1404-1407.
  21. Hitchin N.J., Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 (1995), 30-112.
  22. Hitchin N.J., Hypercomplex manifolds and the space of framings, in The Geometric Universe (Oxford, 1996), Oxford Univ. Press, Oxford, 1998, 9-30.
  23. Maier R.S., Nonlinear differential equations satisfied by certain classical modular forms, Manuscripta Math. 134 (2011), 1-42, arXiv:0807.1081.
  24. Nehari Z., Conformal mapping, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1952.
  25. Ramanujan S., On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159-186.
  26. Ramanujan S., Notebooks, Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957.
  27. Ramanujan S., Collected Papers, Amer. Math. Soc., Providence, RI, 2000.
  28. Randall M., Flat $(2,3,5)$-distributions and Chazy's equations, SIGMA 12 (2016), 029, 28 pages, arXiv:1506.02473.
  29. Randall M., Schwarz triangle functions and duality for certain parameters of the generalised Chazy equation, arXiv:1607.04961.
  30. Rosenhead L. (Editor), Laminar boundary layers, Clarendon Press, Oxford, 1963.
  31. Schwarz H.A., Ueber diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292-335.
  32. Vidūnas R., Algebraic transformations of Gauss hypergeometric functions, Funkcial. Ekvac. 52 (2009), 139-180, math.CA/0408269.

Previous article  Next article   Contents of Volume 13 (2017)