Stability analysis of the spatiotemporal Lugiato-Lefever model for Kerr optical frequency combs in the anomalous and normal dispersion regimes

We propose a detailed stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs in whispering-gallery-mode resonators when they are pumped in either the anomalous- or normal-dispersion regime. We analyze the spatial bifurcation structure of the stationary states depending on two parameters that are experimentally tunable; namely, the pump power and the cavity detuning. Our study demonstrates that, in both the anomalous- and normal-dispersion cases, nontrivial equilibria play an important role in this bifurcation map because their associated eigenvalues undergo critical bifurcations that are actually foreshadowing the existence of localized and extended spatial dissipative structures. The corresponding bifurcation maps are evidence of a considerable richness from a dynamical standpoint. The case of anomalous dispersion is indeed the most interesting from the theoretical point of view because of the considerable variety of dynamical behavior that can be observed. For this case we study the emergence of super- and subcritical Turing patterns (or primary combs) in the system via modulational instability. We determine the areas where bright isolated cavity solitons emerge, and we show that soliton molecules can emerge as well. Very complex temporal patterns can actually be observed in the system, where solitons (or soliton complexes) coexist with or without mutual interactions. Our investigations also unveil the mechanism leading to the phenomenon of breathing solitons. Two routes to chaos in the system are identified; namely, a route via the destabilization of a primary comb, and another via the destabilization of solitons. For the case of normal dispersion, we unveil the mechanism leading to the emergence of weakly stable Turing patterns. We demonstrate that this weak stability is justified by the distribution of stable and unstable fixed points in the parameter space (flat states). We show that dark cavity solitons can emerge in the system, and also show how these solitons can coexist in the resonator as long as they do not interact with each other. We find evidence of breather solitons in this normal dispersion regime as well. The Kerr frequency combs corresponding to all these spatial dissipative structures are analyzed in detail, along with their stability properties. A discussion is led about the possibility to gain unifying comprehension of the observed spectra from the dynamical complexity of the system.