Symmetries of the Continuous and Discrete Krichever–Novikov Equation

Decio Levi ; Pavel Winternitz ; Ravil I. Yamilov

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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 $1\le n\le 5$. The highest dimensions, namely $n=5$ and $n=4$ occur only in the integrable cases.

Détail

Keywords: symmetry classification, integrable PDEs, integrable differential-difference equations.

Bibliographie

[1] Adler V.E., “Bäcklund transformation for the Krichever–Novikov equation”, Int. Math. Res. Not., 1998:1 (1998), 1–4 ; arXiv: solv-int/9707015 | DOI | MR

[2] Adler V.E., Shabat A.B., Yamilov R.I., “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125 (2000), 1603–1661 | DOI | MR | Zbl

[3] Bruzon M.S., Gandarias M.L., “Classical and nonclassical reductions for the Krichever–Novikov equation”, ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics, AIP Conf. Proc., 1281, 2010, 2147–2150 | DOI

[4] Dorodnitsyn V., Applications of Lie groups to difference equations, Differential and Integral Equations and Their Applications, 8, CRC Press, Boca Raton, FL, 2011 | MR

[5] Dubrovin B., Krichever I.M., Novikov S.P., “Integrable systems. I”, Current Problems in Mathematics. Fundamental Directions, 4, VINITI, Moscow, 1985, 179–277 (Russian) | MR | Zbl

[6] Güngör F., Lahno V.I., Zhdanov R.Z., “Symmetry classification of KdV-type nonlinear evolution equations”, J. Math. Phys., 45 (2004), 2280–2313 ; arXiv: nlin.SI/0201063 | DOI | MR

[7] Krichever I.M., Novikov S.P., “Holomorphic bundles over algebraic curves and non-linear equations”, Russ. Math. Surv., 35:6 (1980), 53–79 | DOI | MR | Zbl

[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 | Zbl

[9] Levi D., Petrera M., “Continuous symmetries of the lattice potential KdV equation”, J. Phys. A: Math. Theor., 40 (2007), 4141–4159 ; arXiv: math-ph/0701079 | DOI | MR | Zbl

[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 pp., arXiv: 0802.1850 | DOI | MR | Zbl

[11] Levi D., Winternitz P., “Continuous symmetries of discrete equations”, Phys. Lett. A, 152 (1991), 335–338 | DOI | MR

[12] Levi D., Winternitz P., “Symmetries of discrete dynamical systems”, J. Math. Phys., 37 (1996), 5551–5576 | DOI | MR | Zbl

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

[14] Levi D., Winternitz P., Yamilov R.I., “Lie point symmetries of differential-difference equations”, J. Phys. A: Math. Theor., 43 (2010), 292002, 14 pp., arXiv: 1004.5311 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 pp. ; arXiv: 1011.0070 | DOI | MR | Zbl

[17] Mokhov O.I., “Canonical Hamiltonian representation of the Krichever–Novikov equation”, Math. Notes, 50 (1991), 939–945 | DOI | MR | Zbl

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

[19] Nijhoff F.W., Hone A., Joshi N., “On a Schwarzian PDE associated with the KdV hierarchy”, Phys. Lett. A, 267 (2000), 147–156 ; arXiv: solv-int/9909026 | DOI | MR | Zbl

[20] Novikov D.P., “Algebraic-geometric solutions of the Krichever–Novikov equation”, Theoret. and Math. Phys., 121 (1999), 1567–1573 | DOI | MR | Zbl

[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 | MR | Zbl

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

[23] Sokolov V.V., “On the Hamiltonian property of the Krichever–Novikov equation”, Sov. Math. Dokl., 30 (1984), 44–46 | MR | Zbl

[24] Svinolupov S.I., Sokolov V.V., Yamilov R.I., “On Bäcklund transformations for integrable evolution equations”, Soviet Math. Dokl., 28 (1983), 165–168 | MR | Zbl

[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 | DOI | MR | Zbl

[26] Winternitz P., “Symmetries of discrete systems”, Discrete Integrable System, Lecture Notes in Phys., 644, eds. B. Grammaticos, Y. Kosmann-Schwarzbach and T. Tamizhmani, Springer, Berlin, 2004, 185–243 ; arXiv: nlin.SI/0309058 | MR | Zbl

[27] Xenitidis P.D., “Integrability and symmetries of difference equations: the Adler–Bobenko–Suris case”, Proc. 4th Workshop “Group Analysis of Differential Equations and Integrable Systems”, University of Cyprus, Nicosia, 2009, 226–242 ; arXiv: 0902.3954 | MR | Zbl

[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 pp. ; arXiv: 0903.3152 | DOI | MR | Zbl

[30] Yamilov R.I., “Classification of discrete evolution equations”, Uspekhi Mat. Nauk, 38:6 (1983), 155–156 (Russian) | MR

[31] Yamilov R.I., “Symmetries as integrability criteria for differential difference equations”, J. Phys. A: Math. Gen., 39 (2006), R541–R623 | DOI | MR | Zbl