Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials

In this work we characterize the dynamical instabilities of localized structures exhibited by a recently introduced [ Gelens et al., Phys. Rev. A 75, 063812 (2007)] generalization of the Lugiato-Lefever model that includes a weakly nonlocal response of an intracavity metamaterial. A rich scenario, in which the localized structures exhibit different types of oscillatory instabilities, tristability, and excitability, including a regime of conditional excitability in which the system is bistable, is presented and discussed. Finally, it is shown that the scenario is organized by a pair of Takens-Bogdanov codimension-2 points.