  A standard nonlinear Schr\"{o}dinger equation has been established by using
the reductive perturbation method to investigate the propagation of
electrostatic dust-acoustic waves, and their modulational instability as well
as the formation of localized electrostatic envelope solitons in an electron
depleted unmagnetized dusty plasma system comprising opposite polarity dust
grains and super-thermal positive ions. The relevant physical plasma parameters
(viz., charge, mass, number density of positive and negative dust grains, and
super-thermality of the positive ions, etc.) have rigorous impact to recognize
the stability conditions of dust-acoustic waves. The present study is useful
for understanding the mechanism of the formation of dust-acoustic envelope
solitons associated with dust-acoustic waves in the laboratory and space
environments.
