Amplitude modulation of three-dimensional low frequency solitary waves in a magnetized dusty superthermal plasma

The amplitude modulation of three dimensional (3D) dust ion-acoustic wave (DIAW) packets is studied in a collisionless magnetized plasma with inertial positive ions, superthermal electrons and negatively charged immobile dust grains. By using the reductive perturbation technique, a 3D-nonlinear Schr\"odinger (NLS) equation is derived, which governs the slow modulation of DIAW packets. The latter are found to be stable in the low-frequency $(\omega<\omega_c)$ regime, whereas they are unstable for $\omega>\omega_c$, and the modulational instability (MI) is related to the modulational obliqueness $(\theta)$. Here, $\omega~(\omega_c)$ is the nondimensional wave (ion-cyclotron) frequency. It is shown that the superthermal parameter $\kappa$, the frequency $\omega_c$ as well as the charged dust impurity $(0<\mu<1)$ shift the MI domains around the $\omega-\theta$ plane, where $\mu$ is the ratio of electron to ion number densities. Furthermore, it is found that the decay rate of instability is quenched by the superthermal parameter $\kappa$ with cut-offs at lower wave number of modulation ($K$), however, it can be higher (lower) with increasing values of $\mu$ ($\omega_c$) having cut-offs at higher values of $K$.