Rotating matter waves in Bose–Einstein condensates

In this paper we consider analytically and numerically the dynamics of waves in two-dimensional, magnetically trapped Bose–Einstein condensates in the weak interaction limit. In particular, we consider the existence and stability of azimuthally modulated structures such as rings, multi-poles, soliton necklaces, and vortex necklaces. We show how such structures can be constructed from the linear limit through Lyapunov–Schmidt techniques and continued to the weakly nonlinear regime. Subsequently, we examine their stability, and find that among the above solutions the only one which is always stable is the vortex necklace. The analysis is given for both attractive and repulsive interactions among the condensate atoms. Finally, the analysis is corroborated by numerical bifurcation results, as well as by numerical evolution results that showcase the manifestation of the relevant instabilities.