Electrical structures of interfaces: a Painlevé II model

Abstract

A Painlevé II model derived out of the classical Nernst–Planck system is applied in the context of boundary value problems that describe the electric field distribution in a regionx>0 occupied by an electrolyte. For privileged flux ratios of the ion concentrations, the auto-Bäcklund transformation admitted by the Painlevé II equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems. These representations involve, in turn, either Yablonskii–Vorob’ev polynomials or classical Airy functions. The requirement that the electric field distribution and ion concentrations in these representations be non-singular imposes constraints on the physical parameters. These are investigated in detail along with asymptotic properties.