Spectral stability of vortices in two-dimensional Bose-Einstein condensates via the Evans function and Krein signature

TL;DR

This paper analyzes the spectral stability of vortex solutions in 2D Bose-Einstein condensates using the Evans function and Krein signature, providing rigorous and computational methods to identify stability.

Contribution

It introduces a rigorous approach combining Evans function computation and Krein signature analysis for stability of vortices in BECs, with new continuation results for eigenvalues.

Findings

No unstable eigenvalues found for vortex solutions

Krein signature reduces computational effort in stability analysis

General continuation results for eigenvalues in infinite-dimensional systems

Abstract

We investigate spectral stability of vortex solutions of the Gross-Pitaevskii equation, a mean-field approximation for Bose-Einstein condensates (BEC) in an effectively two-dimensional axisymmetric harmonic trap. We study eigenvalues of the linearization both rigorously and through computation of the Evans function, a sensitive and robust technique whose use we justify mathematically. The absence of unstable eigenvalues is justified a posteriori through use of the Krein signature of purely imaginary eigenvalues, which also can be used to significantly reduce computational effort. In particular, we prove general basic continuation results on Krein signature for finite systems of eigenvalues in infinite-dimensional problems.