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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.


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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
\by I.~M.~Krichever
\paper Analytic theory of difference equations with rational and elliptic coefficients and the Riemann--Hilbert problem
\jour Uspekhi Mat. Nauk
\yr 2004
\vol 59
\issue 6(360)
\pages 111--150
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 6
\pages 1117--1154

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    1. Arinkin D., Borodin A., “Moduli spaces of D-connections and difference Painlevé equations”, Duke Math. J., 134:3 (2006), 515–556  crossref  mathscinet  zmath  isi  elib
    2. Roques J., “Rational classification and confluence of regular singular difference systems”, Ann. Inst. Fourier (Grenoble), 56:6 (2006), 1663–1699  crossref  mathscinet  zmath  isi
    3. V. A. Yurko, “On recovering Sturm–Liouville operators on graphs”, Math. Notes, 79:4 (2006), 572–582  mathnet  crossref  crossref  mathscinet  zmath  zmath  isi  elib  elib
    4. Krichever I.M., Phong Duong Hong, “On the scaling limit of a singular integral operator”, Geom. Dedicata, 132:1 (2008), 121–134  crossref  mathscinet  zmath  isi  elib
    5. Dzhamay A., “On the Lagrangian structure of the discrete isospectral and isomonodromic transformations”, Int. Math. Res. Not. IMRN, 2008, rnn102, 22 pp.  mathscinet  zmath  isi
    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.  crossref  mathscinet  zmath  isi  elib
    7. Arinkin D., Borodin A., “tau-function of discrete isomonodromy transformations and probability”, Compos. Math., 145:3 (2009), 747–772  crossref  mathscinet  zmath  isi  elib
    8. Sauloy J., “Equations in q-differences and holomorphic vector bundles over the elliptic curve C*/q(Z))”, Astérisque, 2009, no. 323, 397–429  mathscinet  zmath  isi
    9. Eric M. Rains, “An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)”, SIGMA, 7 (2011), 088, 24 pp.  mathnet  crossref  mathscinet
    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  crossref  mathscinet  zmath  isi
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    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  mathscinet  isi
    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  crossref  isi  scopus
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