A complete analysis of a classical Poisson-Nernst-Planck model for ionic flow

In this work, we examine the stationary one-dimensional classical Poisson-Nernst-Planck (cPNP) model for ionic flow – a singularly perturbed boundary value problem (BVP). For the case of zero permanent charge, we provide a complete answer concerning the existence and uniqueness of the BVP. The analysis relies on a number of ingredients: a geometric singular perturbation framework for a reduction to a singular BVP, a reduction of the singular BVP to a matrix eigenvalue problem, a relation between the matrix eigenvalues and zeros of a meromorphic function, and an application of the Cauchy Argument Principle for identifying zeros of the meromorphic function. Once the zeros of the meromorphic function in a stripe are determined, an explicit solution of the singular BVP is available. It is expected that this work would be useful for studies of other PNP systems.