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

This article is cited in 24 scientific papers (total in 24 papers)

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


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

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Received: 15.11.2010

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
\by A.~V.~Mikhailov, J.~P.~Wang, P.~Xenitidis
\paper Recursion operators, conservation laws, and integrability conditions for difference equations
\jour TMF
\yr 2011
\vol 167
\issue 1
\pages 23--49
\jour Theoret. and Math. Phys.
\yr 2011
\vol 167
\issue 1
\pages 421--443

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    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.  mathnet  crossref  mathscinet
    2. Mikhailov A.V., Wang J.P., “A new recursion operator for Adler's equation in the Viallet form”, Phys. Lett. A, 375:45 (2011), 3960–3963  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Xenitidis P., “Symmetries and conservation laws of the ABS equations and corresponding differential-difference equations of Volterra type”, J. Phys. A, 44:43 (2011), 435201, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Wang J.P., “Recursion operator of the Narita-Itoh-Bogoyavlensky lattice”, Stud. Appl. Math., 129:3 (2012), 309–327  crossref  mathscinet  zmath  isi  elib  scopus
    6. Demskoi D.K. Viallet C.-M., “Algebraic entropy for semi-discrete equations”, J. Phys. A, 45:35 (2012), 352001, 10 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    7. Garifullin R.N. Yamilov R.I., “Generalized symmetry classification of discrete equations of a class depending on twelve parameters”, J. Phys. A, 45:34 (2012), 345205, 23 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    8. Xenitidis P., Nijhoff F., “Symmetries and conservation laws of lattice Boussinesq equations”, Phys. Lett. A, 376:35 (2012), 2394–2401  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. F. Khanizadeh, A. V. Mikhailov, Jing Ping Wang, “Darboux transformations and recursion operators for differential–difference equations”, Theoret. and Math. Phys., 177:3 (2013), 1606–1654  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. I. T. Habibullin, M. V. Yangubaeva, “Formal diagonalization of a discrete Lax operator and conservation laws and symmetries of dynamical systems”, Theoret. and Math. Phys., 177:3 (2013), 1655–1679  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. Cheng J., Zhang D., “Conservation Laws of Some Lattice Equations”, Front. Math. China, 8:5, SI (2013), 1001–1016  crossref  mathscinet  zmath  isi  elib  scopus
    12. Grant T.J., Hydon P.E., “Characteristics of Conservation Laws for Difference Equations”, Found. Comput. Math., 13:4 (2013), 667–692  crossref  mathscinet  zmath  isi  elib  scopus
    13. Zhang D.-j., Cheng J.-w., Sun Y.-y., “Deriving Conservation Laws for Abs Lattice Equations From Lax pairs”, J. Phys. A-Math. Theor., 46:26 (2013), 265202  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Sergey Ya. Startsev, “Non-Point Invertible Transformations and Integrability of Partial Difference Equations”, SIGMA, 10 (2014), 066, 13 pp.  mathnet  crossref  mathscinet
    15. R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. Scimiterna Ch. Hay M. Levi D., “On the Integrability of a New Lattice Equation Found By Multiple Scale Analysis”, J. Phys. A-Math. Theor., 47:26 (2014), 265204  crossref  mathscinet  zmath  isi  scopus
    17. Mikhailov A.V., Xenitidis P., “Second Order Integrability Conditions For Difference Equations: An Integrable Equation”, Lett. Math. Phys., 104:4 (2014), 431–450  crossref  mathscinet  zmath  adsnasa  isi  scopus
    18. Startsev S.Ya., “Darboux Integrable Discrete Equations Possessing An Autonomous First-Order Integral”, J. Phys. A-Math. Theor., 47:10 (2014), 105204  crossref  mathscinet  zmath  isi  scopus
    19. Demskoi D.K., “Quad-Equations and Auto-Backlund Transformations of NLS-Type Systems”, J. Phys. A-Math. Theor., 47:16 (2014), 165204  crossref  mathscinet  zmath  adsnasa  isi  scopus
    20. Habibullin I.T. Poptsova M.N., “Asymptotic Diagonalization of the Discrete Lax pair Around Singularities and Conservation Laws For Dynamical Systems”, J. Phys. A-Math. Theor., 48:11 (2015), 115203  crossref  mathscinet  zmath  adsnasa  isi  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathscinet  zmath  isi
    23. Lou S. Shi Y. Zhang D.-j., “Spectrum transformation and conservation laws of lattice potential KdV equation”, Front. Math. China, 12:2 (2017), 403–416  crossref  mathscinet  zmath  isi  scopus
    24. Xenitidis P., “Determining the Symmetries of Difference Equations”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 474:2219 (2018), 20180340  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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