Extended and localized Hopf-Turing mixed-mode in non-instantaneous Kerr cavities

We investigate the spatio-temporal dynamics of a ring cavity filled with a non-instantaneous Kerr medium and driven by a coherent injected beam. We show the existence of a stable mixed-mode solution that can be either extended or localized in space. The mixed-mode solutions are obtained in a regime where Turing instability (often called modulational instability) interacts with self-pulsing phenomenon (Andronov-Hopf bifurcation). We numerically describe the transition from stationary inhomogeneous solutions to a branch of mixed-mode solutions. We characterize this transition by constructing the bifurcation diagram associated with these solutions. Finally, we show stable localized mixed-mode solutions, which consist of time-periodic oscillations that are localized in space.