  The modulational instability (MI) criteria of dust-ion-acoustic (DIA) waves
(DIAWs) have been investigated in a four-component pair-ion plasma having
inertial pair-ions, inertialess non-thermal non-extensive electrons, and
immobile negatively charged massive dust grains. A nonlinear Schr\"{o}dinger
equation (NLSE) is derived by using reductive perturbation method. The
nonlinear and dispersive coefficients of the NLSE can predict the
modulationally stable and unstable parametric regimes of DIAWs and associated
first and second order DIA rogue waves (DIARWs). The MI growth rate and the
configuration of the DIARWs are examined, and it is found that the MI growth
rate increases (decreases) with increasing the number density of the negatively
charged dust grains in the presence (absence) of the negative ions. It is also
observed that the amplitude and width of the DIARWs increase (decrease) with
the negative (positive) ion mass. The implications of the results to laboratory
and space plasmas are briefly discussed.
