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

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.