Since the experimental creation of Bose-Einstein Condensates (BEC) in 1995,
BEC are one of the most active areas of modern condensed-matter physics. From the point of view of
nonlinear waves, the interesting phenomena is that the Gross-Pitaevskii equation, which models BEC,
supports the existence of various types of solitary wave solutions, and particularly vortex solutions.
Although there is a large literature on linear stability of vortices, the rigorous and complete approach to this problem is absent.
Using the Evans function we study the a single localized vortex trapped in an harmonic trap.
We confirm that singly-quantized vortices are linearly stable and that the linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability. Moreover, we propose a significant reduction of the numerical cost of the algorithm by replacing the traditional winding number calculation by using the information on the Krein signature of possible unstable eigenvalues. (This is a joint work with Robert L. Pego.)
|