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

SIGMA 4 (2008), 077, 14 pages      arXiv:0802.1850

On Miura Transformations and Volterra-Type Equations Associated with the Adler-Bobenko-Suris Equations

Decio Levi a, Matteo Petrera b, a, Christian Scimiterna b, a and Ravil 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) Dipartimento di Fisica E. Amaldi, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy
c) Ufa Institute of Mathematics, 112 Chernyshevsky Str., Ufa 450077, Russia

Received August 29, 2008, in final form October 30, 2008; Published online November 08, 2008

We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler-Bobenko-Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund transformations for some particular cases of the discrete Krichever-Novikov equation found by Yamilov (YdKN equation). This enables us to construct new generalized symmetries for the ABS equations. The same can be said about the generalizations of the ABS equations introduced by Tongas, Tsoubelis and Xenitidis. All of them generate Bäcklund transformations for the YdKN equation. The higher order generalized symmetries we construct in the present paper confirm their integrability.

Key words: Miura transformations; generalized symmetries; ABS lattice equations.

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  1. Adler V.E., On the structure of the Bäcklund transformations for the relativistic lattices, J. Nonlinear Math. Phys. 7 (2000), 34-56, nlin.SI/0001072.
  2. Adler V.E., Bobenko A.I., Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233 (2003), 513-543, nlin.SI/0202024.
  3. Adler V.E., Bobenko A.I., Suris Yu.B., Discrete nonlinear hyperbolic equations. Classification of integrable cases, arXiv:0705.1663.
  4. Adler V.E., Shabat A.B., Yamilov R.I., The symmetry approach to the integrability problem, Teoret. Mat. Fiz. 125 (2000), 355-424 (English transl.: Theoret. and Math. Phys. 125 (2000), 1603-1661).
  5. Adler V.E., Suris Yu.B., Q4: integrable master equation related to an elliptic curve, Int. Math. Res. Not. 2004 (2004), no. 47, 2523-2553, nlin.SI/0309030.
  6. Adler V.E., Veselov A.P., Cauchy problem for integrable discrete equations on quad-graph, Acta Appl. Math. 84 (2004), 237-262, math-ph/0211054.
  7. Atkinson J., Bäcklund transformations for integrable lattice equations, J. Phys. A: Math. Theor. 41 (2008) 135202, 8 pages, arXiv:0801.1998.
  8. Atkinson J., Hietarinta J., Nijhoff F.W., Seed and soliton solutions for Adler's lattice equation, J. Phys. A: Math. Theor. 40 (2007), F1-F8, nlin.SI/0609044.
  9. Bobenko A.I., Suris Yu.B., Integrable systems on quad-graphs, Int. Math. Res. Not. 2002 (2002), no. 11, 573-611, nlin.SI/0110004.
  10. Case K.M., Kac M., A discrete version of the inverse scattering problem, J. Math. Phys. 14 (1973), 594-603.
  11. Chiu S.C., Ladik J.F., Generating exactly soluble nonlinear discrete evolution equations by a generalized Wronskian technique, J. Math. Phys. 18 (1977), 690-700.
  12. Francoise J.P., Naber G., Tsou S.T. (Editors), Encyclopedia of mathematical physics, Elsevier, 2007.
  13. Galor O., Discrete dynamical systems, Springer, Berlin, 2007.
  14. Hirota R., Nonlinear partial difference equations. I. A difference analog of the Korteweg-de Vries equation, J. Phys. Soc. Japan 43 (1977), 1423-1433.
    Hirota R., Nonlinear partial difference equations. III. Discrete sine-Gordon equation, J. Phys. Soc. Japan 43 (1977), 2079-2086.
  15. Krichever I.M., Novikov S.P., Holomorphic bundles over algebraic curves, and nonlinear equations, Uspekhi Mat. Nauk 35 (1980), no. 6, 47-68 (in Russian).
  16. Levi D., Nonlinear differential-difference equations as Bäcklund transformations, J. Phys. A: Math. Gen. 14 (1981), 1083-1098.
  17. Levi D., Petrera M., Continuous symmetries of the lattice potential KdV equation, J. Phys. A: Math. Theor. 40 (2007), 4141-4159, math-ph/0701079.
  18. Levi D., Petrera M., Scimiterna C., The lattice Schwarzian KdV equation and its symmetries, J. Phys. A: Math. Theor. 40 (2007), 12753-12761, math-ph/0701044.
  19. Levi D., Winternitz P., Continuous symmetries of difference equations, J. Phys. A: Math. Gen. 39 (2006), R1-R63, nlin.SI/0502004.
  20. 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.
  21. Mikhailov A.V., Shabat A.B., Yamilov R.I., The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems, Uspekhi Mat. Nauk 42 (1887), no. 4, 3-53 (English transl.: Russian Math. Surveys 42 (1987), no. 4, 1-63).
  22. Nijhoff F.W., Lax pair for the Adler (lattice Krichever-Novikov) system, Phys. Lett. A 297 (2002), 49-58, nlin.SI/0110027.
  23. Nijhoff F.W., Capel H.W., The discrete Korteweg-de Vries equation, Acta Appl. Math. 39 (1995), 133-158.
  24. Rasin O.G., Hydon P.E., Symmetries of integrable difference equations on the quad-graph, Stud. Appl. Math. 119 (2007), 253-269.
  25. Sandevan J.T., Discrete dynamical systems. Theory and applications, The Clarendon Press, Oxford University Press, New York, 1990.
  26. Shabat A.B., Yamilov R.I., Symmetries of nonlinear chains, Algebra i Analiz 2 (1990), 183-208 (English transl.: Leningrad Math. J. 2 (1991), 377-400).
  27. Tongas A., Tsoubelis D., Papageorgiou V., Symmetries and group invariant reductions of integrable partial difference equations, in Proc. 10th Int. Conf. in Modern Group Analysis (October 24-31, 2004, Larnaca, Cyprus), Editors N.H. Ibragimov, C. Sophocleous and P.A. Damianou, 2004, 222-230.
  28. Tongas A., Tsoubelis D., Xenitidis P., Affine linear and D4 symmetric lattice equations: symmetry analysis and reductions, J. Phys. A: Math. Theor. 40 (2007), 13353-13384, arXiv:0707.3730.
  29. Yamilov R.I., Construction scheme for discrete Miura transformations, J. Phys. A: Math. Gen. 27 (1994), 6839-6851.
  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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