On the solution of the modified Ginzburg-Landau type equation for   one-dimensional superconductor in presence of a normal layer
  We perform an analytical and numerical study of the crossover from the Josephson effect to the bulk superconducting flow for two identical one-dimensional superconductors, co-existing with a layer of normal material.   A generalized Ginzburg-Landau (GL) model, proposed by S.J. Chapman, Q. Du and M.D. Gunzburger was used in modeling the whole structure. When the thickness of the normal layer is very small, the introduction of three effective potentials of specified strength leads to an exact analytical solution of the modified stationary GL equation.   The resulting current density-phase offset relation is analyzed numerically. We show that the critical Josephson current density corresponds to a bifurcation of the solutions of the nonlinear boundary value problem coupled with the modified GL-equation. The influence of the second term in the Fourier-decomposition of the supercurrent density-phase relation is also investigated.   We derive also a simple analytical formula for the critical Josephson current. 
Z.D. Genchev (Bulg. Acad. Sci.) and T.L. Boyadjiev (Univ. of Sofia,   Bulgaria)