B\"acklund flux-quantization in a model of electrodiffusion based on Painlev\'e II

A previously-established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential equations, and has the second-order Painlev\'e II equation at its core. Solutions are now grouped by B\"acklund transformations into infinite sequences, partially labelled by two B\"acklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current-density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck, and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck's charge-neutral boundary-conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary-conditions. In an approximate form, these lead to radiation boundary conditions for Painlev\'e II. Illustrative numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.