Pattern selection and modulational instability in the one-dimensional modified complex Ginzburg-Landau equation

We study analytically modulational instability in the one-dimensional modified complex Ginzburg-Landau equation for the travelling wave systems. The linear stability analysis is used to get domains of instability. We derive the Lange and Newell's criterion for modulational instability. Moreover, it is shown that a modulationally unstable pattern is selected and propagates into an initially unstable motionless state in the system.