Painleve III and Manning's Counterion Condensation

Abstract
Using results of McCoy, Tracy, and Wu [J. Math. Phys. (N.Y.) 18, 1058 (1977)] the Manning counterion condensation for rodlike polyelectrolyte solutions in the presence of salt is rigorously proven. It is demonstrated that the Manning condensation belongs to a special case of the Kosterlitz-Thouless-like phase transition in the presence of boundaries.