Transient ion neutralization by electrons

The nonlinear initial-boundary-value problems describing the lateral neutralization of ion beams for the cases that (i) an auxiliary electric field accelerates the electrons into the ion space, and (ii) the electrons are injected into the ion space at a prescribed current density are treated. Analytical solutions are derived which give the position and speed of the neutralization front as a function of time, and the temporal development of the electron density, velocity, and electric fields during the neutralization process.