Derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation

We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger 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, [equation not represented here], with cutoff length [equation not represented here], where ρ0 is the background condensate density far from the vortex lines and κ is the quantum of circulation.