Differential-difference equations associated with the fractional Lax operators

We study integrable hierarchies associated with spectral problems of the form $P\psi =\lambda Q\psi$ where $P,Q$ are difference operators. The corresponding nonlinear differential-difference equations can be viewed as inhomogeneous generalizations of the Bogoyavlensky type lattices. While the latter turn into the Korteweg–de Vries equation under the continuous limit, the lattices under consideration provide discrete analogs of the Sawada–Kotera and Kaup–Kupershmidt equations. The $r$-matrix formulation and several simplest explicit solutions are presented.