Spectrally Stable Encapsulated Vortices for Nonlinear Schrödinger Equations

A large class of multidimensional nonlinear Schrödinger equations admit localized nonradial standing wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (non-localized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeros of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.