Nonlinear dynamics and Kerr frequency comb formation in lattices of coupled microresonators

Recently, substantial progress has been made in the understanding of microresonators frequency combs based on dissipative Kerr solitons (DKSs). However, most of the studies have focused on the single-resonator level. Coupled resonator systems can open new avenues in dispersion engineering and exhibit unconventional four-wave mixing (FWM) pathways. However, these systems still lack theoretical treatment. Here, starting from general considerations for the N-(spatial) dimensional case, we derive a model for a one-dimensional lattice of microresonators having the form of the two-dimensional Lugiato-Lefever equation (LLE) with a complex dispersion surface. Two fundamentally different dynamical regimes can be identified in this system: elliptic and hyperbolic. Considering both regimes, we investigate Turing patterns, regularized wave collapse, and 2D (i.e., spatio-temporal) DKSs. Extending the system to the Su-Schrieffer-Heeger model, we show that the edge-state dynamics can be approximated by the conventional LLE and demonstrate the edge-bulk interactions initiated by the edge-state DKS.