Modulational instability of electrostatic waves in a magnetized dusty plasma with kappa distributed electrons

The modulational instability of dust-ion acoustic wave (DIAW) and dust-ion cyclotron wave (DICW) is investigated in a magnetized plasma containing static dust particles. The static dust particles can be positively or negatively charged, while ions are taken to be dynamic and inertialess electrons follow the kappa distribution in a magnetized dusty plasma. The nonlinear Schrödinger equation is derived to study the amplitude modulation of obliquely propagating electrostatic waves in a magnetized dusty plasma using the Krylov-Bogoliubov-Mitropolsky method. The dispersive and nonlinear coefficients, i.e., P and Q of nonlinear Schrödinger equation, are obtained which depend on the dust charge concentration, the magnetic field intensity, dust charge polarity, angle of wave propagation, and spectral index kappa for the nonthermal electrons. The modulationally stable and unstable regions of DIAW and DICW are investigated numerically, and the illustration of contour plots of product PQ with wave propagation angle θ and critical wave number kc is also presented. It is found that DIAW and DICW become modulationally unstable at low values of wave number k for negatively charged dust particles in comparison with positively charged dust particles or without dust particles case in a magnetized plasma. The stable region for DIAW, whose wave frequency now depends also on wave propagation angle in a magnetized plasma, is found to be increased in comparison with the unmagnetized plasma case. The observations and existence of positively and negatively charged dust particles in different regions of space plasmas and in laboratory experiments are also pointed out.