Particle Kinetics in Asymmetric Electrostatic Waves

We present analytical expressions for the energy distribution functions 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. Using a suitable model for the potential profile, we find, on quite general grounds, that the the integral solutions of Poisson's equation for both of the trapped electron and ion populations may be reduced to elliptic, rather than Maxwellian, functions. As such, the energy distributions of both particle species are affected by an integrable logarithmic singular term. We show that these singularities are interconnected and that a simple relation exist between the coefficients of the logarithmic term, the values of the jump at the discontinuities and the skew asymmetry of the space distribution of the potential. At variance with earlier work, and in spite of the singularities within the asymmetric solitary waves, these particle distributions correctly reproduce smooth boundary conditions and a smooth space distribution of the solitary wave amplitude in excellent agreement with observations