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SIGMA, 2016, Volume 12, 110, 50 pages (Mi sigma1192)  

This article is cited in 4 scientific papers (total in 4 papers)

Commutation Relations and Discrete Garnier Systems

Christopher M. Ormeroda, Eric M. Rainsb

a University of Maine, Department of Mathemaitcs & Statistics, 5752 Neville Hall, Room 322, Orono, ME 04469, USA
b California Institute of Technology, Mathematics 253-37, Pasadena, CA 91125, USA

Abstract: We present four classes of nonlinear systems which may be considered discrete analogues of the Garnier system. These systems arise as discrete isomonodromic deformations of systems of linear difference equations in which the associated Lax matrices are presented in a factored form. A system of discrete isomonodromic deformations is completely determined by commutation relations between the factors. We also reparameterize these systems in terms of the image and kernel vectors at singular points to obtain a separate birational form. A distinguishing feature of this study is the presence of a symmetry condition on the associated linear problems that only appears as a necessary feature of the Lax pairs for the least degenerate discrete Painlevé equations.

Keywords: integrable systems; difference equations; Lax pairs; discrete isomonodromy.

Funding Agency Grant Number
National Science Foundation DMS-1500806
The work of EMR was partially supported by the National Science Foundation under the grant DMS-1500806.


DOI: https://doi.org/10.3842/SIGMA.2016.110

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Full text: http://www.emis.de/journals/SIGMA/2016/110/
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Bibliographic databases:

ArXiv: 1601.06179
MSC: 39A10; 39A13; 37K15
Received: March 30, 2016; in final form October 30, 2016; Published online November 8, 2016
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Citation: Christopher M. Ormerod, Eric M. Rains, “Commutation Relations and Discrete Garnier Systems”, SIGMA, 12 (2016), 110, 50 pp.

Citation in format AMSBIB
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\by Christopher~M.~Ormerod, Eric~M.~Rains
\paper Commutation Relations and Discrete Garnier Systems
\jour SIGMA
\yr 2016
\vol 12
\papernumber 110
\totalpages 50
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\crossref{https://doi.org/10.3842/SIGMA.2016.110}
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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. Yasuhiko Yamada, “An Elliptic Garnier System from Interpolation”, SIGMA, 13 (2017), 069, 8 pp.  mathnet  crossref
    2. Hidehito Nagao, “A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$”, SIGMA, 13 (2017), 092, 18 pp.  mathnet  crossref
    3. Ch. M. Ormerod, E. M. Rains, “An elliptic Garnier system”, Commun. Math. Phys., 355:2 (2017), 741–766  crossref  mathscinet  zmath  isi
    4. H. Nagao, Ya. Yamada, “Variations of the $q$ -Garnier system”, J. Phys. A-Math. Theor., 51:13 (2018), 135204  crossref  mathscinet  zmath  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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