Stable stationary and breathing holes at the onset of a weakly inverted instability

We show numerically different stable localized structures including stationary holes, moving holes, breathing holes, stationary and moving pulses in the one-dimensional subcritical complex Ginzburg–Landau equation with periodic boundary conditions, and using two classes of initial conditions. The coexistence between different types of stable solutions is summarized in a phase diagram. Stable breathing moving holes as well as breathing nonmoving holes have not been described before for dissipative pattern-forming systems including reaction-diffusion systems.