Parametric instability in a collisional dusty plasma

The present work investigates the parametric instability of parallel propagating circularly polarized Alfv́n (pump) waves in a collisional dusty plasma. It is demonstrated that the relative drift between the charged dust and the electrons and ions gives rise to the Hall effect resulting in the modified pump wave characteristics. Although the linearized fluid equations with periodic coefficients are difficult to solve analytically, it is shown that a linear transformation can remove the periodic dependence. The resulting linearized equations with constant coefficients are used to derive an algebraic dispersion relation. The growth rate of the parametric instability is a sensitive function of the amplitude of the pump wave as well as to the ratio of the pump and the dust-cyclotron frequencies. The instability is insensitive to the plasma-beta parameter. The possible application of the result in the astrophysical context is discussed.