Rigorous and generalized derivation of vortex line dynamics in superfluids and superconductors

We prove rigorously the asymptotic motion equation of a vortex line in a superconductor and a superfluid at small coherence length. In superconductors, the leading order term of the motion equation of a vortex line is dominated by the curvature and the normal direction of the vortex line. In superfluids, the leading order term of the motion equation of a vortex line is determined by the curvature and the binormal direction of the vortex line. Fortunately, the motion equation of a vortex line in a superfluid has the same leading order term as the motion equation of a vortex line in an incompressible fluid at high Reynolds numbers as epsilon = (Reynoldsnumber) -1/2. The method of our proof is more rigorous and generalized than the formal asymptotic analysis in the dynamics of fluid dynamic vortices.