Evolution of Vortex Filaments and Reconnections in the Gross–Pitaevskii Equation and its Approximation by the Binormal Flow Equation

Abstract

The evolution of a vortex line following the binormal flow equation (i.e. with a velocity proportional to the local curvature in the direction of the binormal vector) has been postulated as an approximation for the evolution of vortex filaments in both the Euler system for inviscid incompressible fluids and the Gross–Pitaevskii equation in superfluids. We address the issue of whether this is a suitable approximation or not and its degree of validity by using rigorous mathematical methods and direct numerical simulations. More specifically, we show that as the vortex core thickness goes to zero, the vortex core moves (at leading order and for long periods of time) with a velocity proportional to its local curvature and the binormal vector to the curve. The main idea of our analysis lies in a reformulation of the Gross–Pitaevskii equation in terms of associated velocity and vorticity fields that resemble the Euler system written in terms of vorticity in its weak form. We also present full numerical simulations aimed to compare Gross–Pitaevskii and binormal flow in various physical situations of interest such as the periodic evolution of deformed vortex rings and the reconnection of vortex filaments.