The Lugiato-Lefever equation (LLE) has been extensively studied since its derivation in 1987, when this meanfield
model was introduced to describe nonlinear optical cavities. The LLE was originally derived to describe a
ring cavity or a Fabry-Perot resonator with a transverse spatial extension and partially filled with a nonlinear
medium but it has also been shown to be applicable to other types of cavities, such as fiber resonators and
Depending on the parameters used, the LLE can present a monostable or bistable input-output response
curve. A large number of theoretical studies have been done in the monostable regime, but the bistable regime
has remained widely unexplored. One of the reasons for this was that previous experimental setups were not able
to works in such regimes of the parameter space. Nowadays the possibility of reaching such parameter regimes
experimentally has renewed the interest in the LLE.
In this contribution, we present an in-depth theoretical study of the different dynamical regimes that can
appear in parameter space, focusing on the dynamics of localized solutions, also known as cavity solitons (CSs).
We show that time-periodic oscillations of a 1D CS appear naturally in a broad region of parameter space. More
than this oscillatory regime, which has been recently demonstrated experimentally,1 we theoretically report on
several kinds of chaotic dynamics. We show that the existence of CSs and their dynamics is related with the
spatial dynamics of the system and with the presence of a codimension-2 point known as a Fold-Hopf bifurcation
point. These dynamical regimes can become accessible by using devices such as microresonators, for instance
widely used for creating optical frequency combs.