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 Uspekhi Mat. Nauk, 2004, Volume 59, Issue 6(360), Pages 111–150 (Mi umn798)

This article is cited in 16 scientific papers (total in 16 papers)

Analytic theory of difference equations with rational and elliptic coefficients and the Riemann–Hilbert problem

I. M. Kricheverab

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Columbia University

Abstract: A new approach to the construction of the analytic theory of difference equations with rational and elliptic coefficients is proposed, based on the construction of canonical meromorphic solutions which are analytic along “thick” paths. The concept of these solutions leads to the definition of local monodromies of difference equations. It is shown that, in the continuous limit, these local monodromies converge to monodromy matrices of differential equations. In the elliptic case a new type of isomonodromy transformations changing the periods of elliptic curves is constructed.

DOI: https://doi.org/10.4213/rm798

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English version:
Russian Mathematical Surveys, 2004, 59:6, 1117–1154

Bibliographic databases:

UDC: 517.9
MSC: Primary 30E25, 39A11, 34M35; Secondary 34M50, 34M55, 45E05, 30D99, 32S40, 14H52
Received: 28.06.2004

Citation: I. M. Krichever, “Analytic theory of difference equations with rational and elliptic coefficients and the Riemann–Hilbert problem”, Uspekhi Mat. Nauk, 59:6(360) (2004), 111–150; Russian Math. Surveys, 59:6 (2004), 1117–1154

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Arinkin D., Borodin A., “Moduli spaces of D-connections and difference Painlevé equations”, Duke Math. J., 134:3 (2006), 515–556
2. Roques J., “Rational classification and confluence of regular singular difference systems”, Ann. Inst. Fourier (Grenoble), 56:6 (2006), 1663–1699
3. V. A. Yurko, “On recovering Sturm–Liouville operators on graphs”, Math. Notes, 79:4 (2006), 572–582
4. Krichever I.M., Phong Duong Hong, “On the scaling limit of a singular integral operator”, Geom. Dedicata, 132:1 (2008), 121–134
5. Dzhamay A., “On the Lagrangian structure of the discrete isospectral and isomonodromic transformations”, Int. Math. Res. Not. IMRN, 2008, rnn102, 22 pp.
6. Dzhamay A., “Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painlevé equations”, J. Phys. A, 42:45 (2009), 454008, 10 pp.
7. Arinkin D., Borodin A., “tau-function of discrete isomonodromy transformations and probability”, Compos. Math., 145:3 (2009), 747–772
8. Sauloy J., “Equations in q-differences and holomorphic vector bundles over the elliptic curve C*/q(Z))”, Astérisque, 2009, no. 323, 397–429
9. Eric M. Rains, “An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)”, SIGMA, 7 (2011), 088, 24 pp.
10. Ovsienko V., Schwartz R.E., Tabachnikov S., “Liouville-Arnold Integrability of the Pentagram Map on Closed Polygons”, Duke Math. J., 162:12 (2013), 2149–2196
11. Ormerod Ch.M., “Spectral Curves and Discrete Painlevé Equations”, Algebraic and Analytic Aspects of Integrable Systems and Painlevé Equations, Contemporary Mathematics, 651, eds. Dzhamay A., Maruno K., Ormerod C., Amer Mathematical Soc, 2015, 65–85
12. Gautam S., Laredo V.T., “Yangians, quantum loop algebras, and abelian difference equations”, J. Am. Math. Soc., 29:3 (2016), 775–824
13. Gautam S., Toledano Laredo V., “Meromorphic Tensor Equivalence For Yangians and Quantum Loop Algebras”, Publ. Math. IHES, 2017, no. 125, 267–337
14. Ormerod Ch.M., Rains E.M., “An Elliptic Garnier System”, Commun. Math. Phys., 355:2 (2017), 741–766
15. Nijhoff F., Delice N., “On Elliptic Lax Pairs and Isomonodromic Deformation Systems For Elliptic Lattice Equations in Honour of Professor Noumi For the Occasion of His 60Th Birthday”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 487–525
16. Joshi N., Lustri C.J., Luu S., “Nonlinear Q-Stokes Phenomena For Q-Painleve i”, J. Phys. A-Math. Theor., 52:6 (2019), 065204
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