The particle distributions of asymmetric kinetic electrostatic structures

We give the energy distributions of electrons and ions supporting a steady state electrostatic structure in a collisionless plasma. The electric potential of the structure is skew asymmetrically distributed in space. We show that the jump discontinuous, logarithmically singular electron and ion distributions may be reduced to elliptic integrals. We give the coefficients of the logarithmic terms and the jumps at the discontinuities and we show that they are reciprocally proportional. We calculate bounds for the potential skew asymmetry and show that these bounds are regulated by the boundary conditions of the particle distributions. Despite singularities, our treatment reproduces a smooth space distribution of the potential amplitude and electron and ion distributions that are smooth at one of the boundaries of the electrostatic structure