The cylindrical Poisson-Boltzmann equation. I. Transformations and general solutions

In this paper the cylindrical Poisson-Boltzmann equation in reduced coordinates is transformed into an algebraically nonlinear second order ordinary differential equation, which is a particular case of Painlevé's third equation. The only singularities of solutions to this equation are movable poles of second order. Series expansions are developed which express the general solution locally about points of analyticity and about a pole. Back transformation yields local solutions to the cylindrical Poisson-Boltzmann equation. The solutions throughout an interval generally require analytic continuation of these local expressions. In addition to developing exact solutions, asymptotic properties are analyzed for the case of an isolated cylindrical polyelectrolyte. Further, an alternative technique of solution is sketched in which the cylindrical Poisson-Boltzmann equation is transformed into the sine-Gordon equation, which may then be solved by standard methods to give a one parameter family of solutions. A bibliography of references to the third Painlevé equation is presented. The application of these results to the analysis of the theoretical behavior of cylindrical polyelectrolytes in ionic solutions will be presented in a subsequent contribution.