  A proper theoretical research has been carried out to explore the
modulational instability (MI) conditions of dust-ion-acoustic (DIA) waves
(DIAWs) in a three-component dusty plasma system containing inertialess
$\kappa$-distributed electrons, and inertial warm positive ions and negative
dust grains. The novel nonlinear Schr\"{o}dinger equation (NLSE) has been
derived by employing the reductive perturbation method. The analysis under
consideration demonstrates two types of modes, namely, fast and slow DIA modes.
The dispersion and nonlinear properties of the plasma medium, as well as the MI
conditions of DIAWs and the configuration of the energetic rogue waves (RWs)
associated with DIAWs in the Modulationally unstable regime, have been
rigorously changed by the plasma parameters, namely, charge, mass, temperature,
and number density of the plasma species. The findings of our investigation
will be useful in understanding the criteria for the formation of electrostatic
RWs in both astrophysical environments (viz., Jupiter's magnetosphere, cometary
tails, Earth's mesosphere, Saturn's rings, etc.) and laboratory experiments
(viz., Q-machines and Coulomb-crystal).
