We present a systematic derivation of the Biot-Savart equation from the Nonlinear Schr\"odinger equation, in the limit when the curvature radius of vortex lines and the inter-vortex 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, $$ H= \frac{ \kappa^2}{ 8 \pi}\int_{ |{ \bf s}-{ \bf s}'|>\xi_*} \frac{ d{ \bf s} \cdot d { \bf s}'}{ |{ \bf s}-{ \bf s}'|} \,,$$ with cut-off length $\xi_* \approx 0.3416293 /\sqrt{ \rho_0},$ where $\rho_0$ is the background condensate density far from the vortex lines and $\kappa$ is the quantum of circulation.
Categories: physics.flu-dyn nlin.PS