|
SIGMA 5 (2009), 042, 15 pages arXiv:0811.1796
https://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.
pdf (267 kb)
ps (186 kb)
tex (18 kb)
References
- Arinkin D., Borodin A.,
Moduli spaces of d-connections and difference Painlevé equations,
Duke Math. J. 134 (2006), 515-556, math.AG/0411584.
- Arinkin D., Borodin A., Rains E.,
Talk at the SIDE 8 workshop (June, 2008) and Max Planck Institute for
Mathematics (July, 2008).
- 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.
- Boalch P.,
Quivers and difference Painlevé equations, arXiv:0706.2634.
- 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.
- 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.
- Jimbo M., Sakai H.,
A q-analog of the sixth Painlevé equation,
Lett. Math. Phys. 38 (1996), 145-154.
- 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.
- Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y.,
Cubic pencils and Painlevé Hamiltonians,
Funkcial. Ekvac. 48 (2005), 147-160, nlin.SI/0403009.
- Murata M.,
New expressions for discrete Painlevé equations,
Funkcial. Ekvac. 47 (2004), 291-305, nlin.SI/0304001.
- Rains E.,
An isomonodromy interpretation of the elliptic Painlevé equation. I,
arXiv:0807.0258.
- Sakai H.,
Rational surfaces with affine root systems and geometry of
the Painlevé equations,
Comm. Math. Phys. 220 (2001), 165-221.
- Yamada Y.,
Padé method to Painlevé equations,
Funkcial. Ekvac., to appear.
- 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.
|
|