Advances in Mathematical Physics
Volume 2009 (2009), Article ID 461860, 14 pages
doi:10.1155/2009/461860
Research Article

Generalization of Okamoto's Equation to Arbitrary 2×2 Schlesinger System

Dmitry Korotkin1 and Henning Samtleben2

1Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke West, Montreal QC, H4B 1R6, Canada
2Laboratoire de Physique, Ecole Normale Supérieure de Lyon, Université de Lyon, 46, Allée d'Italie, 69364 Lyon Cedex 07, France

Received 10 June 2009; Accepted 11 September 2009

Academic Editor: Alexander P. Veselov

Copyright © 2009 Dmitry Korotkin and Henning Samtleben. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. L. Schlesinger, “Über eine Klasse von differentialsystemen beliebiger ordnung mit festen kritischen punkten,” Journal für die Reine und Angewandte Mathematik, vol. 141, p. 96, 1912.
  2. M. Jimbo, T. Miwa, and Y. Môri, “Density matrix of an impenetrable bose gas and the fifth Painlevé transcendent,” Physica 1D, pp. 80–158, 1980.
  3. M. Jimbo, T. Miwa, and K. Ueno, “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients,” Physica 2D, vol. 2, no. 2, pp. 306–352, 1981. View at MathSciNet
  4. K. Okamoto, “Isomonodromic deformation and Painlevé equations, and the Garnier system,” Journal of the Faculty of Science, University of Tokyo IA, vol. 33, no. 3, pp. 575–618, 1986. View at MathSciNet
  5. K. Okamoto, “Studies on the Painlevé equations. I. Sixth Painlevé equation pVI,” Annali di Matematica Pura ed Applicata, vol. 146, pp. 337–381, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  6. B. Dubrovin and M. Mazzocco, “Canonical structure and symmetries of the Schlesinger equations,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 289–373, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Mazzocco, “Irregular isomonodromic deformations for Garnier systems and Okamoto's canonical transformations,” Journal of the London Mathematical Society, vol. 70, no. 2, pp. 405–419, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Harnad, “Virasoro generators and bilinear equations for isomonodromic tau functions,” in Isomonodromic Deformations and Applications in Physics, vol. 31 of CRM Proceedings and Lecture Notes, pp. 27–36, American Mathematical Society, Providence, RI, USA, 2002. View at Zentralblatt MATH · View at MathSciNet
  9. M. L. Kontsevich, “Virasoro algebra and Teichmuller spaces,” Functional Analysis and Its Applications, vol. 21, no. 2, pp. 78–79, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  10. D. Korotkin and H. Samtleben, “On the quantization of isomonodromic deformations on the torus,” International Journal of Modern Physics A, vol. 12, p. 2013, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. Kawai, “Isomonodromic deformation of Fuchsian-type projective connections on elliptic curves,” Risk and Insurance Management Society, no. 1022, pp. 53–57, 1997. View at Zentralblatt MATH · View at MathSciNet
  12. A. M. Levin and M. A. Olshanetsky, “Hierarchies of isomonodromic deformations and Hitchin systems,” in Moscow Seminar in Mathematical Physics, vol. 191, pp. 223–262, American Mathematical Society, Providence, RI, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  13. K. Takasaki, “Gaudin model, KZB equation and an isomonodromic problem on the torus,” Letters in Mathematical Physics, vol. 44, no. 2, pp. 143–156, 1998. View at Publisher · View at Google Scholar · View at MathSciNet