A simple class of singular, two-species Vlasov equilibria sustainining nonmonotonic potential distributions

We present new elementary, exact weak singular solutions of the steady state, two species, elecrostatic, one dimensional Vlasov-Poisson equations. The distribution of the hot, finite mass, mobile ions is assumed to be log singular at the position of the electric potential's minimum. We show that the electron energy distributions on opposite sides of this minimum are not equal. This leads to a jump discontinuity of the electron distribution across its separatrix. A simple relation exists between the difference of these two electron distributions and that of the ions. The velocity Fourier transform of the electron singular distribution is smooth and appears as a simple Neumann series. Elementary, finite amplitude profiles of the electric potential result from Poisson equation, which are smoothly, but non monotonically and non symmetrically distributed in space. Two such profiles are given explicitly as appropriate for a non monotonic double layer and for a plasma bounded by a surface. The distributions of both electrons and ions supporting such potential meet smooth and kinetically stable boundary conditions at one plasma boundary. For sufficiently small potential to electron temperature ratios, the non thermal, discontinuous electron distribution resulting at the other plasma boundary is also stable against Landau damped perturbations of the electron distribution.