Article
Ufa Mathematical Journal
Volume 9, Number 3, pp. 158-164
On the integrability of a discrete analogue of the Kaup–Kupershmidt equation
Garifullin R.N., Yamilov R.I.
DOI:10.13108/2017-9-3-158
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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.