Existence and stability of vortex solutions of certain nonlinear Schroedinger equations

The nonlinear Schrodinger equation models a wide variety of different physical phenomena ranging from nonlinear optics, water waves, magnetization of ferromagnets to Bose-Einstein condensates (BEC). The structure of the equation supports existence of topologically non-trivial solutions---vortices. Surprisingly, we demonstrate that the Landau-Lifshitz magnetization equation which is formally also a nonlinear Schrodinger equation does not admit such solutions unless they are stationary. On the other hand, the contrary is true for the Gross-Pitaevskii equation which describes the mean-field approximation of BEC. We investigate stability of vortex solutions by means of a very reliable, sensitive and robust technique---the Evans function. This method, although limited to two dimensions, allows us to study rotating axisymmetric BEC for large particle numbers which can be unattainable by other means. We found a singly-quantized vortex linearly stable. The linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability.