  We report on new types of two-component one-dimensional dark solitons (DSs)
in a model of a dual-core waveguide with normal group-velocity dispersion and
Kerr nonlinearity in both cores, the coupling between which is dispersive too.
In the presence of the dispersive coupling, quiescent DSs supported by the
zero-frequency background are always gray, being stable with the out-of-phase
background, i.e., for opposite signs of the fields in the cores. On the
contrary, the background with a nonzero frequency supports quiescent black
solitons which may be stable for both out- and in-phase backgrounds, if the
dispersive coupling is sufficiently strong. Only DSs supported by the
out-of-phase background admit an extension to the case of nonzero phase
mismatch between the cores.
