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

TL;DR

This paper explores the complex spatio-temporal behaviors in a Kerr cavity driven by a coherent beam, revealing stable mixed-mode solutions that are either extended or localized, arising from the interaction of Turing and Hopf instabilities.

Contribution

It demonstrates the existence and characterization of stable extended and localized mixed-mode solutions in Kerr cavities, highlighting the interplay of Turing and self-pulsing instabilities.

Findings

Existence of stable extended and localized mixed-modes.

Transition from stationary to mixed-mode solutions via bifurcation analysis.

Stable localized mixed-modes with time-periodic oscillations.

Abstract

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.