Nucleation and growth of the superconducting phase in the presence of a current

We study the localized stationary solutions of the one-dimensional time-dependent Ginzburg-Landau equations in the presence of a current. These threshold perturbations separate undercritical perturbations which return to the normal phase from overcritical perturbations which lead to the superconducting phase. Careful numerical work in the small-current limit shows that the amplitude of these solutions is exponentially small in the current; we provide an approximate analysis which captures this behavior. As the current is increased toward the stall current ${J}^{*}$, the width of these solutions diverges, resulting in widely separated normal-superconducting interfaces. We map out numerically the dependence of ${J}^{*}$ on $u$ (a parameter characterizing the material) and use asymptotic analysis to derive the behaviors for large $u$ ${(J}^{*}\ensuremath{\sim}{u}^{\ensuremath{-}1/4})$ and small $u$ $(\stackrel{\ensuremath{\rightarrow}}{J}{J}_{c},$ the critical depairing current), which agree with the numerical work in these regimes. For currents other than ${J}^{*}$ the interface moves, and in this case we study the interface velocity as a function of $u$ and $J$. We find that the velocities are bounded both as $\stackrel{\ensuremath{\rightarrow}}{J}0$ and as $\stackrel{\ensuremath{\rightarrow}}{J}{J}_{c},$ contrary to previous claims.