Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities. (English) Zbl 1479.78019

Parra-Rivas, P.; Knobloch, E.; Gelens, L.; Gomila, D.

IMA J. Appl. Math. 86, No. 5, 856-895 (2021).

Summary: Localized coherent structures can form in externally driven dispersive optical cavities with a Kerr-type non-linearity. Such systems are described by the Lugiato-Lefever (LL) equation, which supports a large variety of dynamical states. Here, we review our current knowledge of the formation, stability and bifurcation structure of localized structures in the one-dimensional LL equation. We do so by focusing on two main regimes of operation: anomalous and normal second-order dispersion. In the anomalous regime, localized patterns are organized in a homoclinic snaking scenario, which is eventually destroyed, leading to a foliated snaking bifurcation structure. In the normal regime, localized structures undergo a different type of bifurcation structure, known as collapsed snaking. The effects of third-order dispersion and various dynamical regimes are also described.

MSC:

78A60	Lasers, masers, optical bistability, nonlinear optics
35B32	Bifurcations in context of PDEs
35Q60	PDEs in connection with optics and electromagnetic theory

Keywords:

bifurcation structure; homoclinic snaking; collapsed snaking; non-linear optics