  In this paper, we consider the concentration property of solutions to the
dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions.
On a spatial domain, it has long been conjectured that such a solution
concentrates near some curve evolving according to the binormal curvature flow,
and conversely, that a curve moving this way can be realized in a suitable
sense by some solution to the dispersive Ginzburg-Landau equation. Some partial
results are known with rather strong symmetry assumptions.
  Our main theorems here provide affirmative answer to both conjectures under
certain small curvature assumption. The results are valid for small but fixed
material parameter in the equation, in contrast to the general practice to take
this parameter to its zero limit. The advantage is that we can retain precise
description of the vortex filament structure. The results hold on a long but
finite time interval, depending on the curvature assumption.
