  A theoretical investigation has been made to understand the mechanism of the
formation of both bright and dark envelope soltions associated with
dust-ion-acoustic waves (DIAWs) propagating in an unmagnetized three component
dusty plasma medium having inertial warm positive ions and negative dust
grains, and inertialess non-thermal Cairns' distributed electrons. A nonlinear
Schr\"{o}dinger equation (NLSE) is derived by employing reductive perturbation
method. The effects of plasma parameters, viz., $\gamma_2$ (the ratio of the
positive ion temperature to electron temperature times the charge state of ion)
and $\nu$ (the ratio of the charge state of negative dust grain to positive
ion) on the modulational instability of DIAW which is governed by NLSE, are
extensively studied. It is found that increasing the value of the ion
(electron) temperature reduces (enhances) the critical wave number ($k_c$). The
results of our present theoretical work may be used to interpret the nonlinear
electrostatic structures which can exist in many astrophysical environments and
laboratory plasmas.
