Integrability and dynamics of the Poisson–Boltzmann equation in simple geometries

Abstract

In this article, we conduct a systematic study of the integrability and dynamics of the PB equation in plane, cylinder and sphere geometries. Specifically, for the PB equation with single species and zero permanent charges, explicit solutions are available in the plane and cylinder geometries, while non-integrability in the sphere geometry is demonstrated by the differential Galoisian approach. Furthermore, we establish a stable manifold result on a noncompact straight-line solution of the PB equation, providing a unified dynamical system perspective on the problem of single charged membrane. Notably, in the case of plane geometry, our results reveal that the concave-upward profile of the differential capacitance curve arises from the interplay between anions and cations present in the electrolyte, as opposed to the saddle structure of the PB equation.