Differential-difference equations associated with the fractional Lax operators

We study integrable hierarchies associated with spectral problems of the form Pψ = λQψ, where PandQ are difference operators. The corresponding nonlinear differentialdifference 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 SawadaKotera and KaupKupershmidt equations. The r-matrix formulation and several of the simplest explicit solutions are presented. © 2011 IOP Publishing Ltd.