The derivative expansion perturbation method is applied to a strongly coupled dusty plasma system consisting of negatively charged dust grains, electrons, and ions. The basic equations are reduced to a nonlinear Schrödinger type equation appropriate for describing the modulated dust acoustic (DA) waves. We have examined the modulation (in) stability and the dependence of the system physical parameters (angular frequency and group velocity) on the polarization force variation. Finally, the extended Poincaré-Lighthill-Kuo technique is employed to investigate the head-on collision (HoC) between two DA dark solitons. The analytical phase shifts and the trajectories of these dark solitons after the collision are derived. The numerical illustrations show that the polarization effect has strong influence on the nature of the phase shifts and the trajectories of the two DA dark solitons after collision.

Cylindrical and spherical amplitude modulations of dust acoustic (DA) solitary wave envelopes in a strongly coupled dusty plasma containing nonthermal distributed ions are studied. Employing a reductive perturbation technique, a modified nonlinear Schrödinger equation including the geometrical effect is derived. The influences of nonthermal ions, polarization force, and the geometries on the modulational instability conditions are analyzed and the possible rogue wave structures are discussed in detail. It is found that the spherical DA waves are more structurally stable to perturbations than the cylindrical ones. Possible applications of these theoretical findings are briefly discussed.

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The amplitude modulation of ion acoustic envelope solitary waves in the presence of Cairns-Gurevich distributed electrons has been investigated. Using a reductive perturbation technique, a modified nonlinear Schrödinger equation has been derived. The modulational instability (MI) and its dependence on the system physical parameters and the combined effects of trapped and nonthermal electrons have been analyzed. It is found that the MI maximum growth rate increases (decreases) as the nonthermality (trapping) parameter increases. The present results could be applicable in explaining the basic features of localized electrostatic disturbance in space observations such as the solar energetic particle flows in interplanetary space and the energetic particle events in the Earth's magnetosphere and also in the laser plasma interaction.

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