Propagation of nonplanar dust-acoustic envelope solitary waves in a two-ion-temperature dusty plasma

The evolution of the cylindrical and spherical dust-acoustic envelope solitary wave (DAESW) in an unmagnetized dusty plasma consisting of negatively charged dust fluid and ions of two different temperatures is investigated. By using the reductive perturbation method, the cylindrical and spherical geometry-modified nonlinear Schrödinger equation (GMNLSE) is obtained. The change of the DAESW amplitude due to the cylindrical and spherical geometry effects is deduced analytically. It is shown that there exist two time ranges. On the other hand, the wave amplitude changes with time τ as ${\mathrm{(\tau }}_0{\mathrm{/\tau )}}^{\text{m/2}}$ when the geometry effect is stronger and as ${\mathrm{(\tau }}_0{\mathrm{/\tau )}}^m$ when the geometry effect is weaker, where ${\tau }_0$ is the initial time, and $\text{m=1}$ (2) refers to the cylindrical (spherical) case. The theoretical results are verified by the numerical calculation for the GMNLSE. The modulational instability of dust-acoustic waves governed by the GMNLSE is also presented. It is shown that the propagation of the DAESW in cylindrical geometry, spherical geometry, and planar one-dimensional geometry are very different. The presence of a second component of ions would modify the nature of the modulation instabilities.