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


SIGMA 5 (2009), 042, 15 pages      arXiv:0811.1796      http://dx.doi.org/10.3842/SIGMA.2009.042
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

A Lax Formalism for the Elliptic Difference Painlevé Equation

Yasuhiko Yamada
Department of Mathematics, Faculty of Science, Kobe University, Hyogo 657-8501, Japan

Received November 20, 2008, in final form March 25, 2009; Published online April 08, 2009

Abstract
A Lax formalism for the elliptic Painlevé equation is presented. The construction is based on the geometry of the curves on P1 × P1 and described in terms of the point configurations.

Key words: elliptic Painlevé equation; Lax formalism; algebraic curves.

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References

  1. Arinkin D., Borodin A., Moduli spaces of d-connections and difference Painlevé equations, Duke Math. J. 134 (2006), 515-556, math.AG/0411584.
  2. Arinkin D., Borodin A., Rains E., Talk at the SIDE 8 workshop (June, 2008) and Max Planck Institute for Mathematics (July, 2008).
  3. Arinkin D., Lysenko S., Isomorphisms between moduli spaces of SL(2)-bundles with connections on P1\{x1,...,x4}, Math. Res. Lett. 4 (1997), 181-190.
    Arinkin D., Lysenko S., On the moduli of SL(2)-bundles with connections on P1\{x1,...,x4}, Internat. Math. Res. Notices 1997 (1997), no. 19, 983-999.
  4. Boalch P., Quivers and difference Painlevé equations, arXiv:0706.2634.
  5. Borodin A., Discrete gap probabilities and discrete Painlevé equations, Duke Math. J. 117 (2003), 489-542, math-ph/0111008.
    Borodin A., Isomonodromy transformations of linear systems of difference equations, Ann. of Math. (2) 160 (2004), 1141-1182, math.CA/0209144.
  6. Grammaticos B., Nijhoff F.W., Ramani A., Discrete Painlevé equations, in The Painlevé Property: One Century Later, Editor R. Conte, CRM Ser. Math. Phys., Springer, New York, 1999, 413-516.
  7. Jimbo M., Sakai H., A q-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154.
  8. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., 10E9 solution to the elliptic Painlevé equation, J. Phys. A: Math. Gen. 36 (2003), L263-L272, nlin.SI/0303032.
  9. Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., Cubic pencils and Painlevé Hamiltonians, Funkcial. Ekvac. 48 (2005), 147-160, nlin.SI/0403009.
  10. Murata M., New expressions for discrete Painlevé equations, Funkcial. Ekvac. 47 (2004), 291-305, nlin.SI/0304001.
  11. Rains E., An isomonodromy interpretation of the elliptic Painlevé equation. I, arXiv:0807.0258.
  12. Sakai H., Rational surfaces with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-221.
  13. Yamada Y., Padé method to Painlevé equations, Funkcial. Ekvac., to appear.
  14. Yamada Y., Talk at the Workshop "Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions", July 21-25, 2008, Max Planck Institute for Mathematics, Bonn, Germany.

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