On integrability of a discrete analogue of Kaup–Kupershmidt equation

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.