  The problem of the one dimensional electro-diffusion of ions in a strong
binary electrolyte is considered. In such a system the solute dissociates
completely into two species of ions with unlike charges. The mathematical
description consists of a diffusion equation for each species augmented by
transport due to a self consistent electrostatic field determined by the
Poisson equation. This mathematical framework also describes other important
problems in physics such as electron and hole diffusion across semi-conductor
junctions and the diffusion of ions in plasmas. If concentrations do not vary
appreciably over distances of the order of the Debye length, the Poisson
equation can be replaced by the condition of local charge neutrality first
introduced by Planck. It can then be shown that both species diffuse at the
same rate with a common diffusivity that is intermediate between that of the
slow and fast species (ambipolar diffusion). Here we derive a more general
theory by exploiting the ratio of Debye length to a characteristic length scale
as a small asymptotic parameter. It is shown that the concentration of either
species may be described by a nonlinear integro-differential equation which
replaces the classical linear equation for ambipolar diffusion but reduces to
it in the appropriate limit. Through numerical integration of the full set of
equations it is shown that this nonlinear equation provides a better
approximation to the exact solution than the linear equation it replaces.
