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The modulational instability of dust-ion-acoustic (DIA) mode and associatedrogue waves in a three component dusty plasma system (containing inertial warmion and negatively charged dust fluids along with inertialess $q$-distributedelectrons) has been theoretically investigated. A nonlinear Schr\"{o}dingerequation (NLSE) has been derived by employing the reductive perturbationmethod. It is observed that the dusty plasma system under considerationsupports the fast and slow DIA modes, and that the dispersion and nonlinearcoefficients of the NLSE determine the parametric regimes not only for themodulationally stable and unstable fast DIA mode, but also for the formation ofthe DIA rogue waves. The parametric regimes for the modulational instability ofthe fast DIA mode, and the criterion for the formation of the DIA rogue waveshave been found to be significantly modified by the effects of the relevantplasma parameters, particularly, mass and charge state of ion and dust species,number density of the plasma species, and non-extensive parameter $q$, etc. Itis found that the modulationally stable parametric regime decreases (increases)with the increase in the value of positive (negative) $q$. The numericalanalysis has also shown that the nonlinearity as well as the amplitude andwidth of the rogue waves increases (decreases) with the mass of positive ion(negative dust grains) while decreases (increases) with the charge state of thepositive ion (negative dust). The applications of our present work in space(viz., Earth ionosphere, magnetosphere, molecular clouds, interstellar medium,cometary tails, and planetary rings, etc.) and laboratory plasmas have beenpinpointed.

Modulational instability of dust-ion-acoustic mode and associated rogue waves in a non-extensive plasma medium