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We develop a general approach to the description of dispersive shock waves(DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin-Onotype dispersion term involving the Hilbert transform. Integrability of thegoverning equation is not a pre-requisite for the application of this methodwhich represents a modification of the DSW fitting method previously developedfor 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 theCalogero-Sutherland dispersive hydrodynamics for which the classification ofall solution types arising from the Riemann step problem is constructed and thekey physical parameters (DSW edge speeds, lead soliton amplitude, intermediateshelf level) of all but one solution type are obtained in terms of the initialstep data. The analytical results are shown to be in excellent agreement withresults of direct numerical simulations.

Dispersive shock waves in systems with nonlocal dispersion of Benjamin-Ono type