Breathing and randomly walking pulses in a semilinear Ginzburg–Landau system

We consider a system consisting of the cubic complex Ginzburg–Landau equation which is linearly coupled to an additional linear equation. The model is known in the context of dual-core nonlinear optical fibers with one active and one passive cores. By means of systematic simulations, we find new types of stable localized excitations, which exist in the system in addition to the earlier found stationary pulses. The new localized excitations include pulses existing on top of a small-amplitude background (that may be regular or chaotic) above the threshold of instability of the zero solution, and breathers into which stationary pulses are transformed through a Hopf bifurcation below the aforementioned threshold. A sharp border between stable stationary pulses and breathers, which precludes their coexistence, is identified. Stable bound states of two breathers with a phase shift 12π between their internal vibrations are found too. Above the threshold, the pulse is standing if the background oscillations are regular; if the background is chaotic, the pulse is randomly walking. With the increase of the system’s size, additional randomly walking pulses are spontaneously generated. The random walk of different pulses in a multi-pulse state is partly synchronized due to their mutual repulsion. At a large overcriticality, the multi-pulse state goes over into a spatiotemporal chaos.