
SIGMA 15 (2019), 062, 15 pages arXiv:1903.11893
https://doi.org/10.3842/SIGMA.2019.062
Integrable Modifications of the ItoNaritaBogoyavlensky Equation
Rustem N. Garifullin and Ravil I. Yamilov
Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, 112 Chernyshevsky Street, Ufa 450008, Russia
Received April 01, 2019, in final form August 14, 2019; Published online August 23, 2019
Abstract
We consider fivepoint differentialdifference equations. Our aim is to find integrable modifications of the ItoNaritaBogoyavlensky equation related to it by noninvertible discrete transformations. We enumerate all modifications associated to transformations of the first, second and third orders. As far as we know, such a classification problem is solved for the first time in the discrete case. We analyze transformations obtained to specify their nature. A number of new integrable fivepoint equations and new transformations have been found. Moreover, we have derived one new completely discrete equation. There are a few nonstandard transformations which are of the Miura type or are linearizable in a nonstandard way. We have also proved that the orders of possible transformations are restricted by the number five in this problem.
Key words: Miura transformation; integrable differentialdifference equation; ItoNaritaBogoyavlensky equation.
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References
 Adler V.E., Necessary integrability conditions for evolutionary lattice equations, Theoret. and Math. Phys. 181 (2014), 13671382, arXiv:1406.1522.
 Adler V.E., Integrable Möbiusinvariant evolutionary lattices of second order, Funct. Anal. Appl. 50 (2016), 257267, arXiv:1605.00018.
 Bogoyavlensky O.I., Integrable discretizations of the KdV equation, Phys. Lett. A 134 (1988), 3438.
 Garifullin R.N., Gubbiotti G., Yamilov R.I., Integrable discrete autonomous quadequations admitting, as generalized symmetries, known fivepoint differentialdifference equations, J. Nonlinear Math. Phys. 26 (2019), 333357, arXiv:1810.11184.
 Garifullin R.N., Yamilov R.I., Integrable discrete nonautonomous quadequations as Bäcklund autotransformations for known Volterra and Toda type semidiscrete equations, J. Phys. Conf. Ser. 621 (2015), 012005, 18 pages, arXiv:1405.1835.
 Garifullin R.N., Yamilov R.I., Levi D., Noninvertible transformations of differentialdifference equations, J. Phys. A: Math. Theor. 49 (2016), 37LT01, 12 pages, arXiv:1604.05634.
 Garifullin R.N., Yamilov R.I., Levi D., Classification of fivepoint differentialdifference equations, J. Phys. A: Math. Theor. 50 (2017), 125201, 27 pages, arXiv:1610.07342.
 Garifullin R.N., Yamilov R.I., Levi D., Classification of fivepoint differentialdifference equations II, J. Phys. A: Math. Theor. 51 (2018), 065204, 16 pages, arXiv:1708.02456.
 Itoh Y., An $H$theorem for a system of competing species, Proc. Japan Acad. 51 (1975), 374379.
 Kuznetsova M.N., Pekcan A., Zhiber A.V., The KleinGordon equation and differential substitutions of the form $v=\phi(u,u_x,u_y)$, SIGMA 8 (2012), 090, 37 pages, arXiv:1111.7255.
 Levi D., Petrera M., Scimiterna C., Yamilov R., On Miura transformations and Volterratype equations associated with the AdlerBobenkoSuris equations, SIGMA 4 (2008), 077, 14 pages, arXiv:0802.1850.
 Mikhailov A.V., Xenitidis P., Second order integrability conditions for difference equations: an integrable equation, Lett. Math. Phys. 104 (2014), 431450, arXiv:1305.4347.
 Narita K., Soliton solution to extended Volterra equation, J. Phys. Soc. Japan 51 (1982), 16821685.
 Papageorgiou V.G., Nijhoff F.W., On some integrable discretetime systems associated with the Bogoyavlensky lattices, Phys. A 228 (1996), 172188.
 Scimiterna C., Hay M., Levi D., On the integrability of a new lattice equation found by multiple scale analysis, J. Phys. A: Math. Theor. 47 (2014), 265204, 16 pages, arXiv:1401.5691.
 Sokolov V.V., On the symmetries of evolution equations, Russian Math. Surveys 43 (1988), no. 5, 165204.
 Startsev S.Ya., On hyperbolic equations that admit differential substitutions, Theoret. and Math. Phys. 127 (2001), 460470.
 Startsev S.Ya., On nonpoint invertible transformations of difference and differentialdifference equations, SIGMA 6 (2010), 092, 14 pages, arXiv:1010.0361.
 Suris Yu.B., The problem of integrable discretization: Hamiltonian approach, Progress in Mathematics, Vol. 219, Birkhäuser Verlag, Basel, 2003.
 Wadati M., Transformation theories for nonlinear discrete systems, Progr. Theoret. Phys. Suppl. 59 (1976), 3663.
 Xenitidis P., Determining the symmetries of difference equations, Proc. A. 474 (2018), 20180340, 20 pages.
 Yamilov R.I., Invertible changes of variables generated by Bäcklund transformations, Theoret. and Math. Phys. 85 (1990), 12691275.
 Yamilov R.I., On the construction of Miura type transformations by others of this kind, Phys. Lett. A 173 (1993), 5357.
 Yamilov R.I., Construction scheme for discrete Miura transformations, J. Phys. A: Math. Gen. 27 (1994), 68396851.
 Yamilov R.I., Symmetries as integrability criteria for differential difference equations, J. Phys. A: Math. Gen. 39 (2006), R541R623.
 Zhang H., Tu G.Z., Oevel W., Fuchssteiner B., Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure, J. Math. Phys. 32 (1991), 19081918.

