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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 158–164 (Mi ufa383)

On integrability of a discrete analogue of Kaup–Kupershmidt equation

R. N. Garifullin, R. I. Yamilov

Institute of Mathematics, Ufa Scientific Center, RAS, Chenryshevsky str. 112, 450008, Ufa, Russia

Abstract: We study a new example of the equation obtained as a result of a recent generalized symmetry classification of differential-difference equations defined on five points of an one-dimensional lattice. We establish that in the continuous limit this new equation turns into the well-known Kaup–Kupershmidt equation. We also prove its integrability by constructing an $L-A$ pair and conservation laws. Moreover, we present a possibly new scheme for constructing conservation laws from $L-A$ pairs.
We show that this new differential-difference equation is similar by its properties to the discrete Sawada–Kotera equation studied earlier. Their continuous limits, namely the Kaup–Kupershmidt and Sawada–Kotera equations, play the main role in the classification of fifth order evolutionary equations made by V. G. Drinfel'd, S. I. Svinolupov and V. V. Sokolov.

Keywords: differential-difference equation, integrability, Lax pair, conservation law.

 Funding Agency Grant Number Russian Science Foundation 15-11-20007 The research is supported by the Russian Science Foundation (project no. 15-11-20007).

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English version:
Ufa Mathematical Journal, 2017, 9:3, 158–164 (PDF, 361 kB); https://doi.org/10.13108/2017-9-3-158

Bibliographic databases:

UDC: 517.9
MSC: 37K10, 35G50, 39A10
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Citation: R. N. Garifullin, R. I. Yamilov, “On integrability of a discrete analogue of Kaup–Kupershmidt equation”, Ufimsk. Mat. Zh., 9:3 (2017), 158–164; Ufa Math. J., 9:3 (2017), 158–164

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. R. N. Garifullin, R. I. Yamilov, D. Levi, “Classification of five-point differential-difference equations II”, J. Phys. A-Math. Theor., 51:6 (2018), 065204
2. R. N. Garifullin, R. I. Yamilov, “Ob integriruemosti reshetochnykh uravnenii s dvumya kontinualnymi predelami”, Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 152, VINITI RAN, M., 2018, 159–164
3. R. N. Garifullin, G. Gubbiotti, I R. Yamilov, “Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations”, J. Nonlinear Math. Phys., 26:3 (2019), 333–357
4. G. Gubbiotti, “Algebraic entropy of a class of five-point differential-difference equations”, Symmetry-Basel, 11:3 (2019), 432
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