Modulational instability of dust ion-acoustic waves and associated envelope solitons in a quantum complex plasma

A two-fluid model is used to investigate the nonlinear amplitude modulation of dust-ion-acoustic waves (DIAWs) propagating in an electron–dust–ion plasma. Electrons are assumed to be inertialess, while ions and dust particles are considered mobile. The quantum effects, i.e., statistical and diffraction effects due to ions and electrons are also incorporated. The fundamental fluid equations are reduced to a nonlinear Schrödinger equation (NLSE) via the reductive perturbation technique. The modulational instability and growth rate of the DIAWs, as determined by the NLSE, are discussed. The dust concentration and quantum-mechanical effects of the ions profoundly influence the conditions for modulational instability. The dark-envelope solitons (black/gray) of localized DIAWs are discussed in detail. It was demonstrated that increasing values of relevant plasma parameters significantly influenced both the amplitude and phase of modulated dark solitons. The results of this study are essential for analyzing nonlinear structures in nanomaterials and nanostructures.