  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.
