  Asymptotic reductions of a defocusing nonlocal nonlinear Schr\"{o}dinger
model in $(3+1)$-dimensions, in both Cartesian and cylindrical geometry, are
presented. First, at an intermediate stage, a Boussinesq equation is derived,
and then its far-field, in the form of a variety of Kadomtsev-Petviashvilli
(KP) equations for right- and left-going waves, is found. KP models include
versions of the KP-I and KP-II equations, in Cartesian and cylindrical
geometry. Solitary waves solutions, planar or ring-shaped, and of dark or
anti-dark type, are also predicted to occur. Their nature and stability is
determined by a parameter defined by the physical parameters of the original
nonlocal system. It is thus found that (dark) anti-dark solitary waves are only
supported by a weak (strong) nonlocality, and are unstable (stable) in
higher-dimensions. Our analytical predictions are corroborated by direct
numerical simulations.
