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 TMF, 2011, Volume 167, Number 1, Pages 23–49 (Mi tmf6624)

Recursion operators, conservation laws, and integrability conditions for difference equations

A. V. Mikhailova, J. P. Wangb, P. Xenitidisa

a Applied Mathematics Department, University of Leeds, UK
b School of Mathematics and Statistics, University of Kent, UK

Abstract: We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler–Bobenko–Suris equations.

Keywords: difference equation, integrability, integrability condition, symmetry, conservation law, recursion operator

DOI: https://doi.org/10.4213/tmf6624

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English version:
Theoretical and Mathematical Physics, 2011, 167:1, 421–443

Bibliographic databases:

Citation: A. V. Mikhailov, J. P. Wang, P. Xenitidis, “Recursion operators, conservation laws, and integrability conditions for difference equations”, TMF, 167:1 (2011), 23–49; Theoret. and Math. Phys., 167:1 (2011), 421–443

Citation in format AMSBIB
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• https://doi.org/10.4213/tmf6624
• http://mi.mathnet.ru/eng/tmf/v167/i1/p23

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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. Decio Levi, Christian Scimiterna, “Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf–Cole Transformations”, SIGMA, 7 (2011), 079, 24 pp.
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4. Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097
5. Wang J.P., “Recursion operator of the Narita-Itoh-Bogoyavlensky lattice”, Stud. Appl. Math., 129:3 (2012), 309–327
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8. Xenitidis P., Nijhoff F., “Symmetries and conservation laws of lattice Boussinesq equations”, Phys. Lett. A, 376:35 (2012), 2394–2401
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21. Mikhailov A.V. Papamikos G. Wang J.P., “Darboux Transformation for the Vector sine-Gordon Equation and Integrable Equations on a Sphere”, Lett. Math. Phys., 106:7 (2016), 973–996
22. Hietarinta J. Joshi N. Nijhoff F., “Discrete Systems and Integrability”, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, Cambridge Univ Press, 2016, 1–445
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24. Xenitidis P., “Determining the Symmetries of Difference Equations”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 474:2219 (2018), 20180340
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