Weakly nonlinear analysis and localized structures in nonlinear cavities with metamaterials

We consider an optical ring cavity filled with a metamaterial and with a Kerr medium. The cavity is driven by a coherent radiation beam. The modelling of this device leads to the well known Lugiato-Lefever equation with high order diffraction. We show that this effect alters in depth the space-time dynamics of this device. A weakly nonlinear analysis in the vicinity of the first threshold associated with the Turing instability is performed. This analysis allows us to determine the parameter regime where transition from super- to sub-critical bifurcation occurs. When the modulational instability appears subcritically, we show that bright localized structures of light may be generated in two-dimensional setting. Close to the second threshold associated with the Turing instability, dark localized structures are generated and their snaking bifurcation diagram is constructed.