Instability criteria and pattern formation in the complex Ginzburg-Landau equation with higher-order terms.

We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings.