Breathing and randomly walking pulses in a semilinear Ginzburg-Landau   system
  A system consisting of the cubic complex Ginzburg-Landau equation which is linearly coupled to an additional linear dissipative equation, is considered. The model was introduced earlier in the context of dual-core nonlinear optical fibers with one active and one passive cores. We argue that it may also possibly describe traveling-wave convection in a channel with an inner vertical partition. 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) {\em above} the threshold of instability of the zero solution, and breathers into which stationary pulses are transformed through a Hopf bifurcation below the zero-solution instability threshold. A sharp border between the stable stationary pulses and breathers, precluding their coexistence, is identified. Stable bound states of two breathers with a phase shift $\pi /2$ between their internal vibrations are found too. Above the threshold, the pulses are standing if the small-amplitude background oscillations are regular; if the background is chaotic, the pulses are randomly walking. With the increase of the system's size, more randomly walking pulses are spontaneously generated. The random walk of different pulses in a multi-pulse state is synchronized (but not completely) due to their mutual repulsion. At a large overcriticality, the multi-pulse state goes over into a spatiotemporal chaos. 
Hidetsugu Sakaguchi and Boris A. Malomed