  We develop a general approach to the description of dispersive shock waves
(DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin-Ono
type dispersion term involving the Hilbert transform. Integrability of the
governing equation is not a pre-requisite for the application of this method
which represents a modification of the DSW fitting method previously developed
for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e.
reducible to the KdV equation in the weakly nonlinear, long wave,
unidirectional approximation). The developed method is applied to the
Calogero-Sutherland dispersive hydrodynamics for which the classification of
all solution types arising from the Riemann step problem is constructed and the
key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate
shelf level) of all but one solution type are obtained in terms of the initial
step data. The analytical results are shown to be in excellent agreement with
results of direct numerical simulations.
