Pattern formation in a complex Swift-Hohenberg equation with phase bistability

We study pattern formation in a complex Swift Hohenberg equation with phase-sensitive (parametric) gain. Such an equation serves as a universal order parameter equation describing the onset of spontaneous oscillations in extended systems submitted to a kind of forcing dubbed rocking when the instability is towards long wavelengths. Applications include two-level lasers and photorefractive oscillators. Under rocking, the original continuous phase symmetry of the system is replaced by a discrete one, so that phase bistability emerges. This leads to the spontaneous formation of phase-locked spatial structures like phase domains and dark-ring (phase-) cavity solitons. Stability of the homogeneous solutions is studied and numerical simulations are made covering all the dynamical regimes of the model, which turn out to be very rich. Formal derivations of the rocked complex Swift-Hohenberg equation, using multiple scale techniques, are given for the two-level laser and the photorefractive oscillator.