Rigorous and Generalized Derivation of Vortex Line Dynamics in Superfluids and Superconductors

Abstract

We prove rigorously the asymptotic motion equation of a vortex line in a superconductor and a superfluid at small coherence length $\epsilon$. 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 $\epsilon = (Reynolds number)^{-\frac{1}{2}}$. The method of our proof is more rigorous and generalized than the formal asymptotic analysis in the dynamics of fluid dynamic vortices.