Azimuthal modulational instability of vortices in the nonlinear Schrodinger equation

摘要

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrodinger (NLS) equation. The method of studying the stability relies on freezing the radial direction in the Lagrangian functional of the NLS in order to form a quasi-one-dimensional azimuthal equation of motion, and then applying a stability analysis in Fourier space of the azimuthal modes. We formulate predictions of growth rates of individual modes and find that vortices are unstable below a critical azimuthal wave number. Steady-state vortex solutions are found by first using a variational approach to obtain an asymptotic analytical ansatz, and then using it as an initial condition to a numerical optimization routine. The stability analysis predictions are corroborated by direct numerical simulations of the NLS. We briefly show how to extend the method to encompass non-local nonlinearities that tend to stabilize such solutions.