Crossover from the Josephson effect to bulk superconducting flow

The crossover between ideal Josephson behavior and uniform superconducting flow is studied by solving exactly the Ginzburg-Landau equation for a one-dimensional superconductor in the presence of an effective δ-function potential of arbitrary strength. As the effective scattering is turned off, the pairs of Josephson solutions with equal current evolve into a uniform and a solitonic solution with nonzero phase offset. It is also argued that a microscopic description of the crossover must satisfy the self-consistency condition, which is shown to guarantee current conservation. The adiabatic response to an external bias is briefly described. The ac Josephson effect is shown to break down when the external voltage is applied at points which are sufficiently far from the junction.