Skip to main content
SpringerLink
Log in

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

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. E. Zakharov, ed., What is Integrability? Springer, Berlin (1991).

    Google Scholar 

  2. A. V. Mikhailov, ed., Integrability (Lect. Notes Phys., Vol. 767), Springer, Berlin (2009).

    MATH  Google Scholar 

  3. P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts Math., Vol. 107), Springer, New York (1993).

    Book  MATH  Google Scholar 

  4. V. V. Sokolov and A. B. Shabat, “Classification of integrable evolution equations,” in: Mathematical Physics Reviews (Sov. Sci. Rev., Sect. C, Vol. 4, S. P. Novikov, ed.), Harwood Academic, Chur (1984), pp. 221–280.

    Google Scholar 

  5. A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Comm. Math. Phys., 115, 1–19 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  6. A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Russ. Math. Surveys, 42, No. 4, 1–63 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  7. A. V. Mikhajlov, A. B. Shabat, and V. V. Sokolov, “The symmetry approach to the classification of integrable equations,” in: What is Integrability? (V. E. Zakharov, ed.), Springer, Berlin (1991), pp. 115–184.

    Google Scholar 

  8. J. A. Sanders and J. P. Wang, J. Differential Equations, 147, 410–434 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. P. Wang, “Symmetries and conservation laws of evolution equations,” Doctoral dissertation, Vrije Universiteit/Thomas Stieltjes Institute, Amsterdam (1998).

    Google Scholar 

  10. V. É. Adler, A. B. Shabat, and R. I. Yamilov, Theor. Math. Phys., 125, 1603–1661 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  11. R. I. Yamilov, Upsekhi Mat. Nauk, 38, No. 6, 155–156 (1983).

    Google Scholar 

  12. R. I. Yamilov, J. Phys. A, 39, R541–R623 (2006).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. F. W. Nijhoff and H. W. Capel, Acta Appl. Math., 39, 133–158 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Grammaticos, R. G. Halburd, A. Ramani, and C.-M. Viallet, J. Phys. A, 42, 454002 (2009).

    Article  MathSciNet  ADS  Google Scholar 

  15. A. I. Bobenko and Yu. B. Suris, Internat. Math. Res. Notices, 2002, 573–611 (2002); arXiv:nlin/0110004v1 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  16. F. W. Nijhoff, Phys. Lett. A, 297, 49–58 (2002); arXiv:nlin/0110027v1 (2001).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. F. W. Nijhoff and A. J. Walker, Glasg. Math. J., 43, No. A, 109–123 (2001).

    Article  MathSciNet  Google Scholar 

  18. V. E. Adler, A. I. Bobenko, and Yu. B. Suris, Comm. Math. Phys., 233, 513–543 (2003); arXiv:nlin/0202024v2 (2002).

    MathSciNet  ADS  MATH  Google Scholar 

  19. V. E. Adler, A. I. Bobenko, and Yu. B. Suris, Funct. Anal. Appl., 43, No. 1, 3–17 (2009).

    Article  MathSciNet  Google Scholar 

  20. M. P. Bellon and C.-M. Viallet, Comm. Math. Phys., 204, 425–437 (1999); arXiv:chao-dyn/9805006v3 (1998).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. C. M. Viallet, Glasg. Math. J., 51 A, 157–163 (2009); arXiv:0802.0294v1 [hep-th] (2008).

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Xenitidis, “Integrability and symmetries of difference equations: The Adler-Bobenko-Suris case,” arXiv: 0902.3954v1 [nlin.SI] (2009).

  23. O. G. Rasin and P. E. Hydon, J. Phys. A, 40, 12763–12773 (2007).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. A. G. Rasin and J. Schiff, J. Phys. A, 42, 175205 (2009); arXiv:0901.0390v1 [nlin.SI] (2009).

    Article  MathSciNet  ADS  Google Scholar 

  25. A. G. Rasin, J. Phys. A, 43, 235201 (2010); arXiv:1001.0724v1 [nlin.SI] (2010).

    Article  MathSciNet  ADS  Google Scholar 

  26. D. Levi and R. I. Yamilov, J. Phys. A, 42, 454012 (2009); arXiv:0902.4421v1 [nlin.SI] (2009).

    Article  MathSciNet  ADS  Google Scholar 

  27. G.-L. Zhang and X.-S. Gao, “Properties of ascending chains for partial difference polynomial systems,” in: Lecture Notes in Artificial Intelligence (Lect. Notes Comput. Sci., Vol. 5081, D. Kapur, ed.), Springer, Berlin (2008), pp. 307–321.

    Google Scholar 

  28. V. G. Papageorgiou, F. W. Nijhoff, and H. W. Capel, Phys. Lett. A, 147, 106–114 (1990).

    Article  MathSciNet  ADS  Google Scholar 

  29. A. Tongas, D. Tsoubelis, and P. Xenitidis, J. Phys. A, 40, 13353–13384 (2007); arXiv:0707.3730v1 [nlin.SI] (2007).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. M. Adler, Invent. Math., 50, 219–248 (1978/79).

    Article  MathSciNet  ADS  Google Scholar 

  32. J. A. Sanders and J. P. Wang, Nonlin. Anal., 47, 5213–5240 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  33. A. B. Shabat and R. I. Yamilov, Phys. Lett. A, 130, 271–275 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  34. A. V. Mikhailov, J. P. Wang, and P. Xenitidis, “Cosymmetries and Nijenhuis recursion operators for difference equations,” arXiv:1009.2403v1 [nlin.SI] (2010).

  35. I. M. Gel’fand and I. Ya. Dorfman, Funct. Anal. Appl., 13, No. 4, 248–262 (1979).

    Article  MathSciNet  Google Scholar 

  36. A. S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento (2), 28, 299–303 (1980).

    Article  MathSciNet  Google Scholar 

  37. I. Ya. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester (1993).

    Google Scholar 

  38. J. P. Wang, J. Math. Phys., 50, 023506 (2009); arXiv:0809.3899v1 [nlin.SI] (2008).

    Article  MathSciNet  ADS  Google Scholar 

  39. J. Hietarinta, J. Nonlin. Math. Phys., 12(Suppl. 2), 223–230 (2005).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics Department, University of Leeds, Leeds, UK

    A. V. Mikhailov & P. Xenitidis

  2. School of Mathematics and Statistics, University of Kent, Kent, UK

    Jing Ping Wang

Authors
  1. A. V. Mikhailov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing Ping Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Xenitidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Mikhailov.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 167, No. 1, pp. 23–49, April, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mikhailov, A.V., Wang, J.P. & Xenitidis, P. Recursion operators, conservation laws, and integrability conditions for difference equations. Theor Math Phys 167, 421–443 (2011). https://doi.org/10.1007/s11232-011-0033-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0033-y

Keywords

Search

Navigation