Responses of the complex Ginzburg–Landau equation under harmonic forcing

We study the effect of external harmonic forcing on a one-dimensional complex Ginzburg–Landau equation (CGLE). For a sufficiently large forcing amplitude, a homogeneous state with no spatial structure is observed. As the forcing amplitude decreases, the state becomes unstable, forming a spatially periodic “stripe” state via a supercritical bifurcation. An approximate phase equation is derived, and an analytic solution for the stripe state is obtained. As the forcing amplitude decreases even further, the system undergoes a depinning transition into the state where the average phase has a non-zero velocity.