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Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

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Mean field models of spatially extended degenerate optical parametric oscillators possess one-dimensional stable domain wall solutions in the presence of diffraction. We characterize these structures as spiral heteroclinic connections and study the spatial frequency of the local oscillations of the signal intensity which distinguish them from diffusion kinks. Close to threshold, at resonance or with positive detunings, the dynamics of two-dimensional diffractive domain walls is ruled by curvature effects with a t(1/2) growth law, and coalescence of domains is observed. In this regime, we show how to stabilize regular and irregular distributions of two-dimensional domain walls by injection of a helical wave at the pump frequency. Further above threshold the shrinking of domains of one phase embedded in the other is stopped by the interaction of the oscillatory tails of the domain walls, leading to cavity solitons surrounded by a characteristic dark ring. We investigate the nature and stability of these localized states, provide evidence of their solitonic character, show that they correspond to spiral homoclinic orbits and find that their threshold of appearance lowers with increasing pump cavity finesse.

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