  A four component space plasma system (consisting of immobile positive ions,
inertial cold positrons as well as hot electrons and positrons following
Cairns' nonthermal distribution function is considered. The nonlinear
propagation of the positron-acoustic (PA) waves, in which the inertia
(restoring force) is provided by the cold positron species (nonthermal pressure
of both hot electron and positron species) has been theoretically investigated
by deriving the nonlinear Schr\"odinger (NLS) equation. It is found from the
numerical analysis of this NLS equation that the space plasma system under
consideration supports the existence of both dark and bright envelope solitons
associated with PA waves, and that the dark (bright) envelope solitons are
modulationally stable (unstable). It is also observed that the basic properties
(viz. stable regime and unstable regime with growth rate) of the PA envelope
solitions are significantly modified by related plasma parameters (viz. number
densities and temperature of plasma species), which correspond to different
realistic space plasma situations.
