Existence and Stability of Vortex Solutions of Certain Nonlinear Schrodinger Equations

Abstract

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. 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. This work justifies previous results in the literature obtained by less reliable methods and opens up a few interesting questions.