Analytic proof linking cavity solitons to stable patterns

Prove that cavity solitons occur only in parameter ranges in which a stable pattern, usually hexagonal in two spatial dimensions, also exists, for all cavity-soliton models under consideration.

Background

The paper presents the “pattern element” interpretation of cavity solitons using a saturable-absorber cavity model. In that example, the phase-plane profile of an isolated cavity soliton closely follows the profile of a spike in a coexisting hexagonal pattern.

The authors state that their experience suggests cavity solitons generally occur only where a stable pattern also exists, but explicitly identify the absence of an analytic proof applicable to all of the cavity-soliton models discussed. The unresolved problem is therefore to establish this relationship mathematically rather than relying on numerical and empirical evidence.