Implications of tristability in dissipative Kerr soliton formation

Dissipative solitons are localized structures (LSs) found in a plethora of different fields of sciences, ranging from plant population ecology to nonlinear optics [1]. In the latter, temporal LSs have been extensively studied in dispersive Kerr cavities such as all-fiber resonators [2]. The formation of these LSs (and their type) is generally related to the coexistence of different states in two different bistable scenarios. The first scenario appears when a periodic state coexists with an uniform one, and consists in a portion of the first embedded in the second. These LSs undergo a bifurcation structure known as standard homoclinic snaking (SHS) [3]. In the second case, two uniform states coexist, and LSs consist in a plateau of one uniform state embedded in another one. These LSs undergo collapsed homoclinic snaking (CHS) [3]. One scenario which has not yet been investigated in nonlinear optics is the so-called tristable regime, where two uniform states coexist with one spatially periodic pattern [4]. This work focus on the impact of tristability on the formation of LSs in Kerr cavities. We show that tristability implies a smooth transition between the SHS and CHS scenarios. To induce tristability, higher order dispersion effects, such as fourth-order dispersion (FOD), must be considered. In this context, the dynamics of the electromagnetic field circulating in these cavities is described by the modified Lugiato-Lefever equation \begin{equation*}\partial_{t}A=-(1+i\Delta)A-id_{2}\partial_{x}^{2}A+id_{4}\partial_{x}^{4}A+i\vert A\vert ^{2}A+S,\end{equation*} where $A$ is the normalized complex electric field amplitude, $t$ represents the slow time coordinate, and $x$ corresponds to the fast time in fiber cavities or angular variable in microresonators. The second-order dispersion and FOD coefficients are respectively $d_{2}$ and $d_{4}$ , that we fix to $d_{2}=-1$ and $d_{4}=1$ . The non-linearity is of Kerr-type, the gain is modeled by $S$ , the losses are linear and $\Delta$ describes the detuning.