Аннотация:
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.
\RBibitem{LevPetSci08}
\by Decio Levi, Matteo Petrera, Christian Scimiterna, Ravil Yamilov
\paper On Miura Transformations and Volterra-Type Equations Associated with the Adler--Bobenko--Suris Equations
\jour SIGMA
\yr 2008
\vol 4
\papernumber 077
\totalpages 14
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\crossref{https://doi.org/10.3842/SIGMA.2008.077}
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Эта публикация цитируется в следующих 27 статьяx:
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Garifullin R.N., Gubbiotti G., Yamilov I R., “Integrable Discrete Autonomous Quad-Equations Admitting, as Generalized Symmetries, Known Five-Point Differential-Difference Equations”, J. Nonlinear Math. Phys., 26:3 (2019), 333–357
Rustem N. Garifullin, Ravil I. Yamilov, “Integrable Modifications of the Ito–Narita–Bogoyavlensky Equation”, SIGMA, 15 (2019), 062, 15 pp.
Vekslerchik V.E., “Solitons of the Vector Kdv and Yamilov Lattices”, J. Phys. A-Math. Theor., 52:46 (2019), 465203
Garifullin R.N., Yamilov R.I., Levi D., “Classification of Five-Point Differential-Difference Equations II”, J. Phys. A-Math. Theor., 51:6 (2018), 065204
В. Э. Адлер, “Интегрируемые семиточечные дискретные уравнения и эволюционные цепочки второго порядка”, ТМФ, 195:1 (2018), 27–43; V. E. Adler, “Integrable seven-point discrete equations and second-order evolution chains”, Theoret. and Math. Phys., 195:1 (2018), 513–528
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Gubbiotti G. Scimiterna C. Levi D., “The Non-Autonomous Ydkn Equation and Generalized Symmetries of Boll Equations”, J. Math. Phys., 58:5 (2017), 053507
Garifullin R.N. Habibullin I.T. Yamilov R.I., “Peculiar Symmetry Structure of Some Known Discrete Nonautonomous Equations”, J. Phys. A-Math. Theor., 48:23 (2015), 235201
Garifullin R.N. Yamilov R.I., “Integrable Discrete Nonautonomous Quad-Equations as Backlund Auto-Transformations For Known Volterra and Toda Type Semidiscrete Equations”, Seventh International Workshop: Group Analysis of Differential Equations and Integrable Systems (Gadeisvii), Journal of Physics Conference Series, 621, IOP Publishing Ltd, 2015, UNSP 012005
Р. Н. Гарифуллин, А. В. Михайлов, Р. И. Ямилов, “Дискретное уравнение на квадратной решетке с нестандартной структурой высших симметрий”, ТМФ, 180:1 (2014), 17–34; R. N. Garifullin, A. V. Mikhailov, R. I. Yamilov, “Discrete equation on a square lattice with a nonstandard structure of generalized symmetries”, Theoret. and Math. Phys., 180:1 (2014), 765–780
Scimiterna Ch., Hay M., Levi D., “on the Integrability of a New Lattice Equation Found By Multiple Scale Analysis”, J. Phys. A-Math. Theor., 47:26 (2014), 265204
Adler V.E., Postnikov V.V., “on Discrete 2D Integrable Equations of Higher Order”, J. Phys. A-Math. Theor., 47:4 (2014), 045206
Zhang, DJ; Cheng, JW; Sun, YY, “Deriving conservation laws for ABS lattice equations from Lax pairs”, Journal of Physics A: Mathematical and Theoretical, 46:26 (2013), 265202
Kassotakis P., Nieszporski M., “On Non-Multiaffine Consistent-Around-the-Cube Lattice Equations”, Phys. Lett. A, 376:45 (2012), 3135–3140
Garifullin R.N. Yamilov R.I., “Generalized Symmetry Classification of Discrete Equations of a Class Depending on Twelve Parameters”, J. Phys. A-Math. Theor., 45:34 (2012), 345205
Levi D., Yamilov R.I., “Generalized symmetry integrability test for discrete equations on the square lattice”, J. Phys. A, 44:14 (2011), 145207, 22 pp.
Mikhailov A.V., Wang J.P., Xenitidis P., “Cosymmetries and Nijenhuis recursion operators for difference equations”, Nonlinearity, 24:7 (2011), 2079–2097
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