An important component of our research is to investigate the stability of the localized optical structures. Each of the models we consider incorporates saturating gain behavior, which leads to a non-local integral-differential linear operator in the linear stability analysis. The effect of the integral contribution has a significant impact on the stability and impedes analytical progress, thus requiring the eigenvalues of the linearized operator to be calculated numerically.
Finite difference (FDA) methods, though easy to implement, converge much too slowly, requiring prohibitively large matrix sizes. Accurate calculations, especially of the zero eigenvalues, are crucial lest we draw false conclusions about the stability. Spectral methods such as the Floquet-Fourier-Hill (FFH) method or methods based on Chebyshev polynomials provide significant improvements in accuracy for a given matrix size — the only trade-off is that the matrices are in general dense.
The two figures on the page show bifurcation diagrams following sech solution branches in the PSA and WGA, where the amplitude is plotted versus the gain. In each case the (spectral) stability was calculated using the FFH method, with solid blue indicating spectral stability and the red dotted curves indicating instability. The WGA exhibits a Hopf bifurcation which is correctly identified using the FFH method. Due to the poor resolution of the zero eigenvalue, the FDA incorrectly indicates instability, which is not observed in numerical simulations.
The four outer panels of the PSA diagram show the results of numerical simulations of the pulse (left) and the eigenvalue spectrum of the linearized operator calculated using the FFH method (right) for the parameters indicated in the bifurcation diagram. The simulation results show the exact sech pulse solution in blue (used also as the initial condition for the simulation), with the resulting simulated pulse after 10,000 cavity round trips shown in red. In each case, the sech pulse is unstable when there are positive eigenvalues, and appears stable when there are not.