  A rigorous theoretical investigation is made to study the characteristics of
dust-acoustic (DA) waves (DAWs) in an electron depleted unmagnetized opposite
polarity dusty plasma system that contains super-thermal ($\kappa$-distributed)
ions, mobile positively and negatively charged dust grains for the first time.
The reductive perturbation method is employed to obtain the NLSE to explore the
modulational instability (MI) conditions for DAWs as well as the formation and
characteristics of gigantic rogue waves. The nonlinear and dispersion
properties of the dusty plasma medium are the prime reasons behind the
formation of rogue waves. The height and thickness of the DARWs associated with
DAWs as well as the MI conditions of DAWs are numerically analyzed by changing
different dusty plasma parameters, such as dust charges, dust and ion number
densities, and ion-temperature, etc. The implications of the results for
various space dusty plasma systems (viz., mesosphere, F-rings of Saturn, and
cometary atmosphere, etc.) as well as laboratory dusty plasma produced by
laser-matter interaction are briefly mentioned.
