Efficient numerical method for linear stability analysis of solitary waves

In this paper we use a numerical relaxation algorithm to improve and generalize the obtainment of the perturbation eigenstates of nonlinear systems. As a model problem we consider the linear stability analysis of the vortex eigenstates of the cubic–quintic nonlinear Schrödinger equation. It is shown by numerical calculations that the relaxation algorithm permits accurate tracing of complex perturbation eigenvalues.

► Relaxation based on Newton method for accurate tracing of complex eigenvalues.
► An initial guess for a non-problematic data can be used as start-up.
► Problematic regions are approached from non-critic ones by continuation technique.