Dark solitons in the Lugiato-Lefever equation with normal dispersion

Abstract

The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.

Coexistence of stable dark- and bright-soliton Kerr combs in normal-dispersion resonators

“…In the absence of TOD, Figure 1(a) shows that a dip in the highintensity HSS A t (red) can evolve into a stable dark soliton (black), while a bump on the low-intensity HSS A b (red) relaxes back to A b . This observation that dark solitons exist, but bright solitons do not, is general [15][16][17]. Reference [16] discussed that dark solitons exist due to the locking of overlapping oscillatory tails in the profile of SWs connecting the upper state A t to the bottom state A b .…”

“…These dark solitons are connected by unstable solution branches that serve to add additional spatial oscillations in their profiles, leading to the broadening of the dark states. This type of bifurcation structure is called collapsed snaking [16,17,41,42], which is significantly different from the homoclinic snaking appearing for dissipative solitons associated with subcritical patterns. Such homoclinic snaking, where many solutions coexist over a fixed parameter range around the Maxwell point, is probably better known and has been widely studied in physics [43,44] and optics [13,[45][46][47].…”

“…In black, the bifurcation diagram of dark solitons is shown for d 3 = 0 and has been discussed in detail in [16,17]. Unstable dark solitons originate from the saddle-node point SN hom,2 and acquire stability at the next turning point when increasing the pump amplitude.…”

“…In contrast to the anomalous regime, dark solitons are found in the normal GVD regime, i.e., low-intensity dips embedded in a highintensity homogeneous background. The bifurcation structure and temporal dynamics of these dark solitons, also called "platicons", have been recently studied [15][16][17], and their origin is intimately related to the locking of switching waves (SWs) connecting coexisting homogeneous state solutions of high and low intensity [16][17][18]. Such dissipative localized structures have also been studied in spatial systems, both in one and two dimensions, with different nonlinearities [19,20].…”

“…(5) is known for d 3 = 0. It has the form A(τ ) − A t = Csech 2 (Bτ ), with C and B depending on the control parameters θ and ρ [17]. We used this solution as an initial guess in the continuation algorithm to track these structures to any value of the parameters θ and ρ (and later also d 3 ) far from SN hom,2 .…”