  We study asymptotic reductions and solitary waves of a weakly nonlocal
defocusing nonlinear Schr\"odinger (NLS) model. The hydrodynamic form of the
latter is analyzed by means of multiscale expansion methods. To the
leading-order of approximation (where only the first moment of the response
function is present), we show that solitary waves, in the form of dark
solitons, are governed by an effective Boussinesq/Benney- Luke (BBL) equation,
which describes bidirectional waves in shallow water. Then, for long times, we
reduce the BBL equation to a pair of Korteweg-de Vries (KdV) equations for
right- and left-going waves, and show that the BBL solitary wave transforms
into a KdV soliton. In addition, to the next order of approximation (where both
the first and second moment of the response function are present), we find that
dark solitons are governed by a higher-order perturbed KdV (pKdV) equation,
which has been used to describe ion-acoustic solitons in plasmas and water
waves in the presence of higherorder effects. The pKdV equation is approximated
by a higher-order integrable system and, as a result, only insubstantial
changes in the soliton shape and velocity are found, while no radiation tails
(in this effective KdV picture) are produced.
