Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials

In this work we characterize the dynamical instabilities of localized structures exhibited by a recently introduced [Gelens et al., Phys. Rev. A 75, 063812 (2007)] generalization of the Lugiato-Lefever model that includes a weakly nonlocal response of an intracavity metamaterial. A rich scenario, in which the localized structures exhibit different types of oscillatory instabilities, tristability, and excitability, including a regime of conditional excitability in which the system is bistable, is presented and discussed. Finally, it is shown that the scenario is organized by a pair of Takens-Bogdanov codimension-2 points.

Figure 1

Phase diagram of DS in the nonlocal Kerr cavity for $\theta =1.23$. DSs do not exist in region I, below the saddle node line (SN). DSs are stable between the saddle-node bifurcation (solid line, SN) and the Hopf bifurcation (dashed line, H), namely, in regions II, III, VI, and VIII (see also Fig. 2). In regions IV and VII delimited by a Hopf and a saddle-loop bifurcation (dotted line, SL), there are stable oscillatory DSs while the stationary DSs are unstable. In region V the static DS is unstable and the phase space generated after the destruction of the limit cycle at the SL induces a regime of excitable DSs. Region VI corresponds to a regime of conditional excitability with both stable and excitable DSs. Finally, in region VIII one encounters tristability: a stationary and an oscillatory DS coexist with the homogeneous solution. Where the saddle-node bifurcation line and the Hopf bifurcation line meet is a Takens-Bogdanov (TB) bifurcation.

Figure 2

Zoom of Fig. 1 near ${\text{TB}}_1$ (a) and ${\text{TB}}_2$ (b). Lines have been determined using the method explained in Sec. 3A with the exception of the SL line in (b). In that case, the filled squares display the location of the SL obtained from numerical integration of Eq. (1), while the gray line through these points is only to guide the eye. $\theta =1.23$.

Figure 3

Qualitative evolution of the phase space at $I_s\approx 0.8$ and $\theta =1.23$ for increasing values of $\beta$, corresponding to the different dynamical behavior in a horizontal cut of Fig. 1. From (a) to (j), one goes from a stable DS (region II) to an oscillating DS (region IV), followed by an excitable DS (region V), a conditional excitable DS (region VI), a coexistence of stable and oscillatory DSs (region VIII), and finally a stable DS (region III).

Figure 4

Bifurcation diagrams showing the maximum intensity of the DS as a function of the background intensity $I_s$, for $\beta =\left(\text{a}\right)0.005$, (b) 0.015, (c) 0.0268, (d) 0.0272, and (e) 0.05. The lowest solid line represents the stable homogeneous solution. The lowest dashed line shows the maximum intensity of the unstable middle branch DS. Above that line, the upper branch DS is shown, where the solid line stands for the stable DS and the dashed line for the unstable DS. In (a), (c), and (d) the system exhibits stable oscillatory behavior (see text). In these cases, the maximum and minimum intensity of the oscillating DS are depicted as filled squares. The gray dashed line, representing the unstable limit cycle, is only a guiding line for the eye. When the limit cycle touches the middle branch DS, a saddle-loop bifurcation occurs. $\theta =1.23$.

Figure 5

Dynamical evolution of the system in the tristable regime, where the spatially homogeneous fundamental branch (not shown), stable oscillatory DS regime (solid line), and stable focus (gray line) coexists.

Figure 6

Qualitative sketch of the phase space at $I_s\approx 0.9$ and $\theta =1.23$ for increasing values of $\beta$, corresponding to a horizontal cut of Fig. 2b. As $\beta$ increases from (a) to (d), one starts with an excitable DS (region V), going through a region of bistability between a stable limit cycle and the stable homogeneous solution (region VII), followed by a coexistence of stable and oscillatory DSs (region VIII), and finally a stable DS (region III).

Figure 7

Spectrum of the middle branch DS at the saddle-loop bifurcation for (a) $\beta =0.0268$ and $I_s=0.84$ and (b) $\beta =0.0294$ and $I_s=0.671$. The filled black circles are the eigenvalues corresponding to the modes with a localized spatial profile. $\theta =1.23$.

Figure 8

The conditional excitability regime is illustrated for $\beta =0.0272$ and $I_s=0.8$. The solid line (excitable trajectory) corresponds to time evolution of the maximum of the peak after a localized perturbation of the homogeneous state that brings the system beyond the excitability threshold but outside the basin of attraction of the upper branch DS. The gray line corresponds to a more strongly localized perturbation that brings the system inside the boundary of attraction of the upper branch DS.

Figure 9

Location $\left({\theta }_{\text{TB}},{\beta }_{\text{TB}}\right)$ of the two TB points as the detuning is increased. The solid and dashed lines correspond to the ${\text{TB}}_1$ and ${\text{TB}}_2$ points, respectively.