Geometric Singular Perturbation Approach to Steady-State Poisson--Nernst--Planck Systems

Boundary value problems of a one-dimensional steady-state Poisson--Nernst--Planck (PNP) system for ion flow through a narrow membrane channel are studied. By assuming the ratio of the Debye length to a characteristic length to be small, the PNP system can be viewed as a singularly perturbed problem with multiple time scales and is analyzed using the newly developed geometric singular perturbation theory. Within the framework of dynamical systems, the global behavior is first studied in terms of limiting fast and slow systems. It is rather surprising that a complete set of integrals is discovered for the (nonlinear) limiting fast system. This allows a detailed description of the boundary layers for the problem. The slow system itself turns out to be a singularly perturbed one, too, which indicates that the singularly perturbed PNP system has three different time scales. A singular orbit (zeroth order approximation) of the boundary value problem is identified based on the dynamics of limiting fast and slow systems. An application of the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of the boundary value problem.