Derivation of the Biot-Savart equation from the nonlinear Schrodinger equation

We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrodinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines, H = kappa(2)/8 pi integral vertical bar s - s vertical bar' >xi(*) ds.ds'/vertical bar s - s' vertical bar, with cutoff length xi(*) approximate to 0.341 6293/root rho(0), where rho(0) is the background condensate density far from the vortex lines and kappa is the quantum of circulation.