  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* ~ u^-1/4)
and small u (J -> J_c, the critical deparing 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 J -> 0 and as J -> J_c,
contrary to previous claims.
