Modulational instability of dust envelope waves with grain and charge distribution

A reasonable normalization for a dusty plasma with many different species of dust grains is adopted. By applying a reductive perturbation technique to the equations governing a dusty plasma with N different species of dust grains, a nonlinear Schrödinger equation (NSLE) is derived that governs the modulation of dust-acoustic waves. The effect of dust size and charge distribution on the modulational instability of these waves is studied. If there are positively charged dust grains, which is a possibility suggested by experimental results, the envelope soliton solutions to the NLSE may be different from the ones associated with a dusty plasma containing mono-sized dust grains. The instability properties may also be different. In the former case the instability region depends on the percentage of electrons residing on the dust grains. In particular, if the number of electrons residing on the dust grains is small enough, the envelope waves are unstable.