Wigner transform and stability of kinetic envelope soliton in an electron depleted dusty plasma with two temperature non-thermal ions

A nonlinear theory of amplitude modulation of electrostatic envelopes in an electron depleted dusty plasma consisting of negatively charged dust grains and two temperature non-thermal ions developed using Vlasov-Poisson equation. By using the Krylov-Bogoliubov-Mitropolsky method a nonlinear Schrodinger equation is derived. The dispersion relation is written in terms of moments of plasma dispersion function 'Z (zeta)' is analyzed numerically. Spatially periodic unstable modes are studied with the help of Wigner transform defining the spectral density at the initial time. This is actually a statistical approach to modulational stability very relevant due to kinetic formulation of our problem. In this method one can study various form of initial spectral distribution and their stability leading to the various conditions on plasma parameters. Our standard modulational stability is the special case of such a treatment. The explicit solution of the nonlinear Schrodinger equation is seen to be different depending upon the sign of 'P' and 'Q' (the dispersive and nonlinear coefficient). In one case we get the standard envelope soliton and the other one leads to either to envelope hole or envelope shock solution. It is very interesting to notice that the non-thermal behavior of the ions have a controlling nature on the modulational stability of the system. The advantage of working with the Wigner transformation is that it can be used as a tool for homogenization, that is in a situation with large number of waves. This type of analysis of modulational instability are now-a-days very important because the quantum nature of plasma is gaining importance.