Explicit solutions of the one-dimensional Vlasov-Poisson system with infinite mass

A collisionless plasma is modelled by the Vlasov-Poisson system in one dimension. A fixed background of positive charge, dependent only upon velocity, is assumed and the situation in which the mobile negative ions balance the positive charge as |x| (right arrow) (infinity)is considered. Thus, the total positive charge and the total negative charge are infinite. In this paper, the charge density of the system is shown to be compactly supported. More importantly, both the electric field and the number density are determined explicitly for large values of |x|.