Generalize the necking‑bifurcation‑mediated transition to higher spatial dimensions

Determine how the transition between standard homoclinic snaking and collapsed homoclinic snaking of localized states, mediated by codimension‑two necking bifurcations in the one‑dimensional Lugiato–Lefever equation with second‑ and fourth‑order dispersion (uniform–pattern–uniform tristable regime), extends to higher spatial dimensions. Specifically, ascertain whether analogous transitions occur for radially symmetric localized states in two and three dimensions by homotopic continuation in system dimension, and characterize the resulting bifurcation structures and state morphologies.

Background

The paper establishes, in a non-variational mean-field optical cavity model (the Lugiato–Lefever equation with fourth-order dispersion), that the reorganization of localized-state bifurcation structures in uniform–pattern–uniform tristability is mediated by a cascade of codimension‑two necking bifurcations. This mechanism yields a transition from standard homoclinic snaking (pattern–uniform locking) to collapsed homoclinic snaking (uniform–uniform locking), and is shown to be generic in one-dimensional extended dissipative systems.

While the one-dimensional case is analyzed in depth using path-continuation and spectral stability, the authors highlight that the generalization of this scenario to higher extended dimensions (e.g., radially symmetric states in 2D and 3D) remains unresolved. They suggest homotopic continuation in system dimension as a possible route, noting that related studies exist for other equations and settings but that the specific transition uncovered here has not been characterized in higher dimensions.