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

Abstract

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 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, $$H=\frac{{\kappa }^2}{8\pi }{\int }_{|\mathbf{s}-{\mathbf{s}}^{\prime }|>{\xi }_{\ast }}\frac{d\mathbf{s}\cdot d{\mathbf{s}}^{\prime }}{|\mathbf{s}-{\mathbf{s}}^{\prime }|},$$ with cut-off length ${\xi }_{\ast }\approx 0.3416293/\sqrt{{\rho }_0},$ where ${\rho }_0$ is the background condensate density far from the vortex lines and $\kappa$ is the quantum of circulation.

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