Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 097, 16 pages      arXiv:1110.5021      http://dx.doi.org/10.3842/SIGMA.2011.097
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Symmetries of the Continuous and Discrete Krichever-Novikov Equation

Decio Levi a, Pavel Winternitz b and Ravil I. Yamilov c
a) Dipartimento di Ingegneria Elettronica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
b) Centre de recherches mathématiques and Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, succ. Centre-ville, H3C 3J7, Montréal (Québec), Canada
c) Ufa Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russian Federation

Received June 16, 2011, in final form October 15, 2011; Published online October 23, 2011

Abstract
A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension n of the Lie point symmetry algebra satisfies 1≤n≤5. The highest dimensions, namely n=5 and n=4 occur only in the integrable cases.

Key words: symmetry classification; integrable PDEs; integrable differential-difference equations.

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References

  1. Adler V.E., Bäcklund transformation for the Krichever-Novikov equation, Int. Math. Res. Not. 1998 (1998), no. 1, 1-4, solv-int/9707015.
  2. Adler V.E., Shabat A.B., Yamilov R.I., Symmetry approach to the integrability problem, Theoret. and Math. Phys. 125 (2000), 1603-1661.
  3. Bruzon M.S., Gandarias M.L., Classical and nonclassical reductions for the Krichever-Novikov equation, in ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics, AIP Conf. Proc. 1281 (2010), 2147-2150.
  4. Dorodnitsyn V., Applications of Lie groups to difference equations, Differential and Integral Equations and Their Applications, Vol. 8, CRC Press, Boca Raton, FL, 2011.
  5. Dubrovin B., Krichever I.M., Novikov S.P., Integrable systems. I, Current Problems in Mathematics. Fundamental Directions, Vol. 4, VINITI, Moscow, 1985, 179-284 (in Russian).
  6. Güngör F., Lahno V.I., Zhdanov R.Z., Symmetry classification of KdV-type nonlinear evolution equations, J. Math. Phys. 45 (2004), 2280-2313, nlin.SI/0201063.
  7. Krichever I.M., Novikov S.P., Holomorphic bundles over algebraic curves and non-linear equations, Russ. Math. Surv. 35 (1980), no. 6, 53-80.
  8. Krichever I.M., Novikov S.P., Holomorphic fiberings and nonlinear equations. Finite zone solutions of rank 2, Sov. Math. Dokl. 20 (1979) 650-654.
  9. Levi D., Petrera M., Continuous symmetries of the lattice potential KdV equation, J. Phys. A: Math. Theor. 40 (2007), 4141-4159, math-ph/0701079.
  10. Levi D., Petrera M., Scimiterna C., Yamilov R.I., On Miura transformations and Volterra-type equations associated with the Adler-Bobenko-Suris equations, SIGMA 4 (2008), 077, 14 pages, arXiv:0802.1850.
  11. Levi D., Winternitz P., Continuous symmetries of discrete equations, Phys. Lett. A 152 (1991), 335-338.
  12. Levi D., Winternitz P., Symmetries of discrete dynamical systems, J. Math. Phys. 37 (1996), 5551-5576.
  13. Levi D., Winternitz P., Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006), R1-R63, nlin.SI/0502004.
  14. Levi D., Winternitz P., Yamilov R.I., Lie point symmetries of differential-difference equations, J. Phys. A: Math. Theor. 43 (2010), 292002, 14 pages, arXiv:1004.5311.
  15. Levi D., Yamilov R.I., Conditions for the existence of higher symmetries of evolutionary equations on the lattice, J. Math. Phys. 38 (1997), 6648-6674.
  16. Levi D., Yamilov R.I., Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011), 145207, 22 pages, arXiv:1011.0070.
  17. Mokhov O.I., Canonical Hamiltonian representation of the Krichever-Novikov equation, Math. Notes 50 (1991), 939-945.
  18. Nijhoff F.W., Capel H., The discrete Korteweg-de Vries equation, Acta Appl. Math. 39 (1995), 133-158.
  19. Nijhoff F.W., Hone A., Joshi N., On a Schwarzian PDE associated with the KdV hierarchy, Phys. Lett. A 267 (2000), 147-156, solv-int/9909026.
  20. Novikov D.P., Algebraic-geometric solutions of the Krichever-Novikov equation, Theoret. and Math. Phys 121 (1999), 1567-1573.
  21. Novikov S.P., Manakov S.V., Pitaevsky L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Contemporary Soviet Mathematics, Plenum, New York, 1984.
  22. Rasin O.G., Hydon P.E., Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119 (2007), 253-269.
  23. Sokolov V.V., On the Hamiltonian property of the Krichever-Novikov equation, Sov. Math. Dokl. 30 (1984), 44-46.
  24. Svinolupov S.I., Sokolov V.V., Yamilov R.I., On Bäcklund transformations for integrable evolution equations, Soviet Math. Dokl. 28 (1983), 165-168.
  25. Weiss J., The Painlevé property for partial differential equations. II. Bäcklund transformation, Lax pairs, and the Schwarzian derivative, J. Math. Phys. 24 (1983), 1405-1413.
  26. Winternitz P., Symmetries of discrete systems, in Discrete Integrable System, Editors B. Grammaticos, Y. Kosmann-Schwarzbach and T. Tamizhmani, Lecture Notes in Phys., Vol. 644, Springer, Berlin, 2004, 185-243, nlin.SI/0309058.
  27. Xenitidis P.D., Integrability and symmetries of difference equations: the Adler-Bobenko-Suris case, in Proc. 4th Workshop "Group Analysis of Differential Equations and Integrable Systems", University of Cyprus, Nicosia, 2009, 226-242, arXiv:0902.3954.
  28. Xenitidis P.D., Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type, arXiv:1105.4779.
  29. Xenitidis P.D., Papageorgiou V.G., Symmetries and integrability of discrete equations defined on a black-white lattice, J. Phys. A: Math. Theor. 42 (2009), 454025, 13 pages, arXiv:0903.3152.
  30. Yamilov R.I., Classification of discrete evolution equations, Uspekhi Mat. Nauk 38 (1983), no. 6, 155-156 (in Russian).
  31. Yamilov R.I., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541-R623.

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