Dynamics of Pattern Formation and Modulational Instabilities in the Quintic Complex Ginzburg–Landau Equation*

We study analytically modulational instability in an extended media, taking the one-dimensional quintic complex Ginzburg–Landau equation model as an example. We show that the quintic term can affect the conditions for modulational instability (MI), and the propagating regime. Numerical simulations demonstrate the validity of analytical predictions. The model presents a rich variety of patterns propagating into the system. The main feature of the travelling shock waves is the propagation of a secondary modulated wave at the same time with a stable plane wave.