Differential–difference equations associated with the fractional Lax operators

We study integrable hierarchies associated with spectral problems of the form P ψ = λ Q ψ, where P and 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 of the simplest explicit solutions are presented.