Synthesis and regularization of the statistical distribution functions of the particles inside an asymmetric solitary wave

We present new singular solutions of the steady state, two species Vlasov-Poisson equations. The hot, finite mass, mobile ions have an energy distribution which is log singular at the position of the electric potential's minimum. We show that a simple relation exists between the unequal electron distributions on opposite sides of this minimum and the ion distribution. The distributions of both species are given in terms of elementary functions and they are subject to smooth boundary conditions at one plasma end. Simple, 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 solutions are investigated in detail as appropriate for a plasma of semi-infinite extent bounded by a surface, and for solitary waves.