Azimuthal Modulational Instability of Vortex Solutions to the Two Dimensional Nonlinear Schrödinger Equation
Ships in 3-5 business days
We study the azimuthal modulational instability (MI) 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 nonlinear equation
numerical optimization routine. The stability analysis predictions are corroborated by direct
numerical simulations of the NLS performed on a polar coordinate finite-difference scheme.