Wave Propagation and Diffusive Transition of Oscillations in Pair   Plasmas with Dust Impurities
  In view of applications to electron-positron pair-plasmas and fullerene pair-ion-plasmas containing charged dust impurities a thorough discussion is given of three-component Plasmas. Space-time responses of multi-component linearized Vlasov plasmas on the basis of multiple integral equations are invoked. An initial-value problem for Vlasov-Poisson -Ampere equations is reduced to the one multiple integral equation and the solution is expressed in terms of forcing function and its space-time convolution with the resolvent kernel. The forcing function is responsible for the initial disturbance and the resolvent is responsible for the equilibrium velocity distributions of plasma species. By use of resolvent equations, time-reversibility, space-reflexivity and the other symmetries are revealed. The symmetries carry on physical properties of Vlasov pair plasmas, e.g., conservation laws. Properly choosing equilibrium distributions for dusty pair plasmas, we can reduce the resolvent equation to: (i) the undamped dispersive wave equations, (ii) wave-diffusive transport equation (iii) and diffusive transport equations of oscillations. In the last case we have to do with anomalous diffusion employing fractional derivatives in time and space. Fractional diffusion equations account for typical anomalous features, which are observed in many systems, e.g. in the case of dispersive transport in amorphous semiconductors, liquid crystals, polymers, proteins and biosystems. 
Barbara Atamaniuk and Andrzej J. Turski