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

Decio Levi ; Matteo Petrera ; Christian Scimiterna ; Ravil Yamilov

Symmetry, integrability and geometry: methods and applications, Tome 4 (2008)

Résumé

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.

Détail

Keywords: Miura transformations; generalized symmetries; ABS lattice equations.

Bibliographie

[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 | DOI | MR | Zbl

[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 | MR | Zbl

[3] Adler V. E., Bobenko A. I., Suris Yu. B., Discrete nonlinear hyperbolic equations. Classification of integrable cases, arXiv:0705.1663 | MR

[4] Theoret. and Math. Phys., 125 (2000), 1603–1661 | DOI | MR | Zbl

[5] Adler V.E., Suris Yu. B., “$\mathrm Q_4$: integrable master equation related to an elliptic curve”, Int. Math. Res. Not., 2004:47 (2004), 2523–2553 ; nlin.SI/0309030 | DOI | MR | Zbl

[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 | MR | Zbl

[7] Atkinson J., “Bäcklund transformations for integrable lattice equations”, J. Phys. A: Math. Theor., 41 (2008), 135202, 8 pp., ages ; arXiv:0801.1998 | DOI | MR | Zbl

[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 | DOI | Zbl

[9] Bobenko A. I., Suris Yu. B., “Integrable systems on quad-graphs”, Int. Math. Res. Not., 2002:11 (2002), 573–611 ; nlin.SI/0110004 | DOI | MR | Zbl

[10] Case K. M., Kac M., “A discrete version of the inverse scattering problem”, J. Math. Phys., 14 (1973), 594–603 | DOI | MR

[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 | DOI | MR | Zbl

[12] Francoise J. P., Naber G., Tsou S. T. (ed.), Encyclopedia of mathematical physics, Elsevier, 2007 | Zbl

[13] Galor O., Discrete dynamical systems, Springer, Berlin, 2007 | MR | Zbl

[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 | MR

[15] Krichever I. M., Novikov S. P., “Holomorphic bundles over algebraic curves, and nonlinear equations”, Uspekhi Mat. Nauk, 35:6 (1980), 47–68 (in Russian) | MR | Zbl

[16] Levi D., “Nonlinear differential-difference equations as Bäcklund transformations”, J. Phys. A: Math. Gen., 14 (1981), 1083–1098 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[19] Levi D., Winternitz P., “Continuous symmetries of difference equations”, J. Phys. A: Math. Gen., 39 (2006), R1–R63 ; nlin.SI/0502004 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[21] Russian Math. Surveys, 42:4 (1987), 1–63 | DOI | MR

[22] Nijhoff F. W., “Lax pair for the Adler (lattice Krichever–Novikov) system”, Phys. Lett. A, 297 (2002), 49–58 ; nlin.SI/0110027 | DOI | MR | Zbl

[23] Nijhoff F. W., Capel H. W., “The discrete Korteweg–de Vries equation”, Acta Appl. Math., 39 (1995), 133–158 | DOI | MR | Zbl

[24] Rasin O. G., Hydon P. E., “Symmetries of integrable difference equations on the quad-graph”, Stud. Appl. Math., 119 (2007), 253–269 | DOI | MR

[25] Sandevan J. T., Discrete dynamical systems. Theory and applications, The Clarendon Press, Oxford University Press, New York, 1990 | MR

[26] Leningrad Math. J., 2 (1991), 377–400 | MR | Zbl

[27] Tongas A., Tsoubelis D., Papageorgiou V., “Symmetries and group invariant reductions of integrable partial difference equations”, Proc. 10th Int. Conf. in Modern Group Analysis (October 24–31, 2004, Larnaca, Cyprus), eds. N. H. Ibragimov, C. Sophocleous and P. A. Damianou, 2004, 222–230

[28] Tongas A., Tsoubelis D., Xenitidis P., “Affine linear and $D_4$ symmetric lattice equations: symmetry analysis and reductions”, J. Phys. A: Math. Theor., 40 (2007), 13353–13384 ; arXiv:0707.3730 | DOI | MR | Zbl

[29] Yamilov R. I., “Construction scheme for discrete Miura transformations”, J. Phys. A: Math. Gen., 27 (1994), 6839–6851 | DOI | MR | Zbl

[30] Yamilov R. I., “Classification of discrete evolution equations”, Uspekhi Mat. Nauk, 38:6 (1983), 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 | DOI | MR | Zbl