Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.

Near a zero-dispersion wavelength, high-order dispersion effects play a central role in a photonic crystal fiber cavity. The study of such effects reveals the existence of stable dissipative dark solitons in the anomalous dispersion regime. Such dark localized structures are not present without high-order dispersion. They consist of dips in the profile of the intensity field. The number of dips and their temporal distribution are determined by the initial conditions. A snaking bifurcation diagram associated with these solutions is constructed.

Taking up to fourth-order dispersion effects into account, we show that fiber resonators become stable for a large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for an arbitrary small but nonzero fourth-order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and open the way to experimental implementation.

We investigate the influence of the stimulated Raman scattering on the formation of bright and dark localized states in all-fiber resonators subject to a coherent optical injection, when operating in the normal dispersion regime. In the absence of the Raman effect, and far from any modulational instability, localized structures form due to the locking of domain walls connecting two coexisting continuous wave states, and undergo a particular bifurcation structure known as collapsed snaking. The stimulated Raman scattering breaks the reflection symmetry of the system, and modifies the dynamics, stability, and locking of domain walls. This modification leads to the formation of, not only dark, but also bright moving localized states, which otherwise are absent. We perform a detailed bifurcation analysis of these localized states, and classify their dynamics and stability as a function of the main parameters of the system.

We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D) localized structures consist of either an isolated light bullet (LB), bound together, or could occur in clusters forming well-defined 3D patterns. They can be seen as stationary states in the reference frame moving with the group velocity of light within the cavity. The number of LBs and their distribution in 3D settings are determined by the initial conditions, while their maximum peak power remains constant for a fixed value of the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation of homoclinic snaking for dissipative light bullets. However, when the strength of the injected beam is increased, LBs lose their stability and the cavity field exhibits giant, short-living 3D pulses. The statistical characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of extreme events, often called rogue waves.