Non-linear electrostatic plasma waves

A solution of the one-dimensional time-independent Vlasov-Poisson system, including the condition of no net current, is constructed for a large class of electric potential functions φ(x), including periodic and solitary waves, as well as monotonic transitions. Displaced Maxwell distributions for the free particles, and an unshifted Maxwell distribution for the trapped ions, are used. The condition of positiveness of the unknown distribution function for the trapped electrons (which can be split into two parts, one depending solely on the chosen wave, the other on the given distributions) is examined. It is shown, for a non-linear wave

that this condition plays no rôle when the thickness l of the wave considerably exceeds the Debye length λ_D(l λ_D). If, however, l lies in the neighbourhood of λ_D(l λ_D), there exist limits for the free parameters. The smallest thickness l_min results if the Mach number M lies in the range 1 > M > 2 and the electronion—temperature ratio θ = T_e/T_i exceeds 10. This confirms the view that the wave is a steepened ion-acoustic wave. l_min decreases with decreasing amplitude _max of the wave and decreasing number of trapped ions, but does not lie below the Debye length as long as the wave is non-linear. In the linear case, the condition of positiveness imposes no restrictions.