  The Lugiato-Lefever equation is a cubic nonlinear Schr\"odinger equation,
including damping, detuning and driving, which arises as a model in nonlinear
optics. We study the existence of stationary waves which are found as solutions
of a four-dimensional reversible dynamical system in which the evolutionary
variable is the space variable. Relying upon tools from bifurcation theory and
normal forms theory, we discuss the codimension 1 bifurcations. We prove the
existence of various types of steady solutions, including spatially localized,
periodic, or quasi-periodic solutions.

Analysis of PDEs