  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.
