Azimuthal modulational instability of vortices in the nonlinear Schrödinger equation

Abstract

We study the azimuthal modulational instability of vortices with different topological charges, in the focusing two-dimensional nonlinear Schrödinger (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 nonlocal nonlinearities that tend to stabilize solutions.

Study of Vortex Ring Dynamics in the Nonlinear Schrödinger Equation Utilizing GPU-Accelerated High-Order Compact Numerical Integrators

“…However, as in the case of bright two-dimensional vortices [13], bright vortex rings are unstable and undergo collapse. As such, their study is not as physically relevant as the dark vortex rings, but are worth a brief exploration (especially since such rings realized in the cubic-quintic NLSE (CQNLS), may be stable for some parameter ranges as is the case with two-dimensional bright vortices in the CQNLS [14]).…”

“…Using a variational approach with asymptotic assumptions, an extremely close analytical approximation to the vortex radial profile is found to be [13] …”

“…For example, the study of modulational instability of vortices in the cubic NLSE in Ref. [13] laid the framework for the study in Ref. [14] of the more complicated stability and dynamics of vortices in the cubic-quintic NLSE, which has closer direct relevance to application (in that case, nonlinear optics).…”

“…(2.18) is more accurate for larger values of the charge m, and is less accurate over smaller values of r for low charge vortices (see Ref. [13] for details).…”

“…(2.18) very accurately approximates the radius and peak value of the vortex profile. The radius of the bright vortex is proportional to the magnitude of the charge (|m|) and is approximated as [13] …”