Evolution of vortex filaments and reconnections in the Gross-Pitaevski equation and its approximation by the binormal flow equation

TL;DR

This paper rigorously investigates the approximation of vortex filament evolution in superfluids by the binormal flow equation, combining mathematical analysis and numerical simulations to assess its validity and limitations.

Contribution

It provides a rigorous mathematical validation and numerical comparison of the binormal flow approximation for vortex filament dynamics in the Gross-Pitaevskii equation.

Findings

Vortex core moves with velocity proportional to local curvature and binormal vector as core thickness approaches zero.

The binormal flow is a valid approximation for vortex filament evolution over long timescales.

Numerical simulations show good agreement between Gross-Pitaevskii dynamics and binormal flow in various scenarios.

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-Pitaevski 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-Pitaevski 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-Pitaevski and binormal flow in various physical situations of interest such as the periodic evolution of deformed vortex rings and the reconnection of vortex filaments.