Geodesic
A Lax Formalism for the Elliptic Difference Painlevé Equation
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A Lax formalism for the elliptic Painlevé equation is presented. The construction is based on the geometry of the curves on $\mathbb P^1\times\mathbb P^1$ and described in terms of the point configurations.
Mots-clés : elliptic Painlevé equation; Lax formalism; algebraic curves. 
@article{SIGMA_2009_5_a41,
     author = {Yasuhiko Yamada},
     title = {A~Lax {Formalism} for the {Elliptic} {Difference} {Painlev\'e} {Equation}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a41/}
}
TY  - JOUR
AU  - Yasuhiko Yamada
TI  - A Lax Formalism for the Elliptic Difference Painlevé Equation
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2009
VL  - 5
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a41/
LA  - en
ID  - SIGMA_2009_5_a41
ER  - 
%0 Journal Article
%A Yasuhiko Yamada
%T A Lax Formalism for the Elliptic Difference Painlevé Equation
%J Symmetry, integrability and geometry: methods and applications
%D 2009
%V 5
%U http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a41/
%G en
%F SIGMA_2009_5_a41
Yasuhiko Yamada. A Lax Formalism for the Elliptic Difference Painlevé Equation. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a41/

[1] Arinkin D., Borodin A., “Moduli spaces of $d$-connections and difference Painlevé equations”, Duke Math. J., 134 (2006), 515–556 ; math.AG/0411584 | DOI | MR | Zbl

[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 $\mathbb P^1\setminus\{x_1,\dots,x_4\}$”, Math. Res. Lett., 4 (1997), 181–190 | MR | Zbl

[4] Boalch P., Quivers and difference Painlevé equations, arXiv:0706.2634 | MR

[5] Borodin A., “Discrete gap probabilities and discrete Painlevé equations”, Duke Math. J., 117 (2003), 489–542 ; ; Borodin A., “Isomonodromy transformations of linear systems of difference equations”, Ann. of Math. (2), 160 (2004), 1141–1182 ; math-ph/0111008math.CA/0209144 | DOI | MR | Zbl | DOI | MR

[6] Grammaticos B., Nijhoff F. W., Ramani A., “Discrete Painlevé equations”, The Painlevé Property: One Century Later, CRM Ser. Math. Phys., ed. R. Conte, Springer, New York, 1999, 413–516 | MR | Zbl

[7] Jimbo M., Sakai H., “A $q$-analog of the sixth Painlevé equation”, Lett. Math. Phys., 38 (1996), 145–154 | DOI | MR | Zbl

[8] Kajiwara K., Masuda T., Noumi M., Ohta Y., Yamada Y., “${}_{10}E_9$ solution to the elliptic Painlevé equation”, J. Phys. A: Math. Gen., 36 (2003), L263–L272 ; nlin.SI/0303032 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[10] Murata M., “New expressions for discrete Painlevé equations”, Funkcial. Ekvac., 47 (2004), 291–305 ; nlin.SI/0304001 | DOI | MR | Zbl

[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 | DOI | MR

[13] Yamada Y., “Padé method to Painlevé equations”, Funkcial. Ekvac., 52:1 (2009), 83–92 | DOI | MR | Zbl

[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, Germanyee