Abstract
The nonlinear interaction between broadband Langmuir waves and electrostatic dust–acoustic perturbations in a uniform collisional dusty plasma is considered, taking into account the combined effects of the Langmuir wave ponderomotive force and the electron Joule heating nonlinearity. This coupling is governed by a Liouville equation for a Langmuir wave packet and an equation for a dust–acoustic perturbation that is driven by the ponderomotive and thermal forces of the random phase Langmuir waves. The mode coupling equations are then Fourier decomposed to derive the nonlinear dispersion relation. The latter is analyzed numerically for a Gaussian Langmuir wave spectrum, by taking parameters that are relevant for low-temperature dusty plasma discharges. It is found that a broadband spectrum of Langmuir waves is subjected to a rapidly growing modulational instability in a dusty plasma.