Derivation of the Biot-Savart equation from the Nonlinear Schr\"odinger equation

TL;DR

This paper systematically derives the classical Biot-Savart law for vortex dynamics from the quantum Nonlinear Schrödinger equation, establishing a Hamiltonian framework with a specific cutoff length.

Contribution

It provides a rigorous derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation in the vortex limit, including the Hamiltonian formulation with a precise cutoff.

Findings

Derived the Biot-Savart equation from the Nonlinear Schrödinger equation.

Established the Hamiltonian form of vortex dynamics.

Identified the cutoff length scale related to condensate density.

Abstract

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.