  Dust-acoustic (DA) waves (DAWs) and their modulational instability (MI) have
been investigated theoretically in a plasma system consisting of inertial
opposite polarity (positively and negatively) warm adiabatic charged dust
particles as well as inertialess non-extensive $q$-distributed electrons and
non-thermal ions. A nonlinear Schr\"{o}dinger (NLS) equation is derived by
using the reductive perturbation method. It has been observed from the analysis
of NLS equaion that the modulationally stable solitary DAWs give rise to the
existence of dark envelope solitons, and that the modulationally unstable
solitary DAWs give rise to the existence of bright envelope solitons or rogue
structures. It is also observed for the fast mode of DAWs that the basic
features (viz. stability of the DAWs, MI growth rate, amplitude and width of
the DA rogue waves, etc.) are significantly modified by the related plasma
parameters (viz. dust masses, dust charge state, non-extensive parameter $q$,
and non-thermal parameter $\alpha$). The results of our present investigation
might be useful for understanding different nonlinear electrostatic phenomena
in both space (viz. ionosphere and mesosphere) and laboratory plasmas (viz.
high intensity laser irradiation and hot cathode discharge).
