Electrophoretic traveling waves

An existence-uniqueness-approximability theory is given for a prototypical mathematical model for the separation of ions in solution by an imposed electric field. The separation is accomplished during the formation of a traveling wave, and the mathematical problem consists in finding a traveling wave solution of a set of diffusion-advection equations coupled to a Poisson equation. A basic small parameter ε ε appears in an apparently singular manner, in that when ε = 0 ε = 0 (which amounts to assuming the solution is everywhere electrically neutral), the last (Poisson) equation loses its derivative, and becomes an algebraic relation among the concentrations. Since this relation does not involve the function whose derivative is lost, the type of "singular" perturbation represented here is nonstandard. Nevertheless, the traveling wave solution depends in a regular manner on ε ε , even at ε = 0 ε = 0 ; and one of the principal aims of the paper is to show this regular dependence.