Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion

The origin, stability, and bifurcation structure of different types of bright localized structures described by the Lugiato-Lefever equation are studied. This mean field model describes the nonlinear dynamics of light circulating in fiber cavities and microresonators. In the case of anomalous group velocity dispersion and low values of the intracavity phase detuning these bright states are organized in a homoclinic snaking bifurcation structure. We describe how this bifurcation structure is destroyed when the detuning is increased across a critical value, and determine how a bifurcation structure known as foliated snaking emerges.

Figure 1

Possible solutions of Eq. (2): (a) HSS solution, (b) pattern state, (c) front or domain wall between a HSS and a pattern, (d) LP solution with several peaks, (e) LP solution with a single peak, and (f) single-peak LS in the form of a spike with monotonic tails. The left panels show the field $U\left(x\right)$, while the right panels show the corresponding solution of the spatial dynamical system (6) projected on the variables $y_1$ and $y_3$. The correspondence is as follows: (a) fixed point or equilibrium, (b) limit cycle, (c) heteroclinic orbit, (d) and (e) multiround and single-round wild homoclinic orbits, and (f) tame homoclinic orbit.

Figure 2

Sketch of the possible organization of the spatial eigenvalues $\lambda$ satisfying Eq. (9) for a spatially reversible system. The terminology corresponding to the different regions is explained in Table 1.

Figure 3

Homogeneous steady states (HSS) of Eq. (2) and their spatial eigenvalue configurations for several values of $\theta$. (a) $\theta =1.5<\sqrt{3}$; (b) $\sqrt{3}<\theta =1.75<2$; (c) $\theta =2$; (d) $\theta =4$. Here, RTBH stands for the reversible Takens-Bogdanov-Hopf bifurcation, RTB for the reversible Takens-Bogdanov bifurcation and HH for the Hamiltonian-Hopf bifurcation, while BD is the Belyakov-Devaney transition and QZ the quadruple zero codimension-two point. Solid (broken) line indicates temporally stable (unstable) HSS.

Figure 4

Comparison between the asymptotic solutions (black line) in Eqs. (27) and (37) and the corresponding numerically exact solutions (red line) obtained by numerical continuation. Both lines are almost indistinguishable on the scale of the figure although the maxima of the asymptotic solution are slightly lower in (a) and higher in (b). (a) LP with $\theta =1.5$ and $\rho =1.11753$. (b) A single spike LS for $\theta =2.5$ and $\rho =1.80460$. In both cases, the distance from the instability threshold is $I_{c,b}-I_0=0.036$.

Figure 5

The parameter plane $(\theta ,\rho )$. The red line corresponds to a HH bifurcation for $\theta <2$ but a BD transition for $\theta >2$. The black lines correspond to the saddle nodes of the HSS $A_0$: ${\mathrm{SN}}_b$ and ${\mathrm{SN}}_t$. In terms of spatial dynamics these lines correspond to the following bifurcations: ${\mathrm{SN}}_t$ is always a RTBH bifurcation; for $\theta <2,{\mathrm{SN}}_b$ is also a RTBH bifurcation, but becomes a RTB bifurcation for $\theta >2$. The saddle nodes of a single peak LS are shown in blue and labeled ${\mathrm{SN}}_1^{\ell }$ and ${\mathrm{SN}}_1^r$; the region in-between is therefore the region of existence of LSs. The purple line represents a (temporal) Hopf bifurcation (H). The inset shows a closeup view around the QZ point.

Figure 6

Homoclinic snaking bifurcation diagram for $\theta =1.5$ and $L=160$. In (a) we plot the diagram using ${\left|\right|A\left|\right|}^2$ while in (b) we subtracted the HSS state $A_0^b$. Solid (dashed) lines indicate stable (unstable) states. (c) (Bottom) Closeup view of the bifurcation diagram in (b) showing the snakes-and-ladders structure: a pair of intertwined branches of $\phi =0$ and $\pi$ LP states (labeled ${\mathrm{\Gamma }}_0, {\mathrm{\Gamma }}_{\pi }$ and shown in blue and green, respectively) with crosslinks consisting of branches of asymmetric states created through a series of pitchfork bifurcations ${\mathrm{P}}_i$. (c) (Top) Drift speed of the rung states as a function of $\rho$ corresponding to the crosslinks ${\mathrm{P}}_i{\mathrm{P}}_{i+1}$ in the snakes-and-ladders diagram. The drift speed vanishes at the pitchfork bifurcations ${\mathrm{P}}_i, i=1,...,4$, and is positive or negative depending on the side at which the extra peak is nucleated as indicated in the $U\left(x\right), V\left(x\right)$ profiles along the bottom of the figure.

Figure 7

(a) Bifurcation diagram for $\theta =2.5$ showing the norm ${\left|\right|A\left|\right|}^2$ as a function of $\rho$. (b) The same diagram but in terms of $\left|\right|A-A_0^b{\left|\right|}^2$ revealing the foliated snaking nature of the diagram; the numbered dots correspond to the profiles shown in the panels on the right. On the real line, the different branches of equispaced multipeak states all bifurcate from the RTB point at ${\mathrm{SN}}_b$. All branches turn around in the vicinity of ${\rho }_b\approx 1.80501$ where the small peaks first appear as $\rho$ decreases. As shown by the boxed peaks in profiles (vi) and (viii), the secondary peaks that appear along ${\mathrm{\Gamma }}_{12}^u$ are identical to the small peaks along the lower branch ${\mathrm{\Gamma }}_1^u$ and similarly for the other branches. These secondary peaks grow with decreasing $\rho$ allowing the branch to terminate on a branch with twice as many peaks in a 2:1 spatial resonance located near the saddle node ${\mathrm{SN}}_{2n}$. The yellow dots, corresponding to profiles (xiii) and (xiv), show that branches in which the small and large peaks are ordered differently are also present [compare these profiles with (viii) and (ix)]. The corresponding branches are indistinguishable in (b) since the quantity shown, $\left|\right|A-A_0^b{\left|\right|}^2$, is insensitive to the ordering of the peaks. In (b) solid (dashed) lines represent stable (unstable) branches. Stability is also indicated using the superscripts $s,u$.

Figure 8

(a), (b) Detail of the saddle node ${\mathrm{SN}}_1^r$. (c) Real part of the temporal eigenvalue $\sigma$ associated to this fold. The zero eigenvalue is a consequence of translation invariance.

Figure 9

(a) Branches of equispaced three- and six-peak states (blue curves) corresponding to the states shown in panels (i)–(vi); as in Fig. 7, branches corresponding to states in which the small and large peaks are ordered differently coincide with ${\mathrm{\Gamma }}_{36}^u$. (b) Branch of equispaced five-peak states (green curve) associated with states (vii) and (viii). In both panels, the gray branches in the background represent those already shown in Fig. 7. In (a), solid (dashed) lines represent stable (unstable) branches.

Figure 10

Schematic bifurcation diagram showing the topological features of foliated snaking. The red branches correspond to Fig. 7 while the blue ones correspond to Fig. 9. The vertical line represents the HSS. On the real line, all branches bifurcate from the RTB bifurcation at ${\mathrm{SN}}_b$ generating a large multiplicity of stationary localized states, only some of which are shown in the figure. This scheme is inspired by a similar one appearing in Ref. [55].

Figure 11

Branches of LP states for $\theta =2.5$ consisting of clumps of peaks. (a) LP branches connected by homotopy with those undergoing homoclinic snaking at $\theta =1.5$ (green and blue curves). The solution branches corresponding to the foliated snaking shown in Fig. 7 are shown in red. (b)–(d) LP states consisting of two-, three-, and four-peak clumps, respectively. The panels show the separation $\mathrm{\Delta }$ between peaks within the clumps as a function of $\rho$ along these branches. The separation $\mathrm{\Delta }$ diverges as $\rho \to {\rho }_{\mathrm{BD}}$ (dashed vertical lines), and at this point a global bifurcation takes place. The clumped states fall on top of the evenly spaced states in the bifurcation diagram [red curves in (a)] because the norm $\left|\right|A-A_0^b{\left|\right|}^2$ only counts the number of peaks.

Figure 12

(a) Schematic unfolding of the global bifurcation occurring in the BD transition. At this bifurcation, a single-peak LP with oscillatory tails changes into a single-peak LS with monotonic tails (i.e., a spike) as $\rho$ increases. As this point is approached from the left, the wavelength of the pattern involved in the formation of the LP states diverges. (b) Schematic unfolding of a single-peak LS in an infinite system (red line) into a large number of branches corresponding to states with different numbers of equispaced peaks. The LPs emerging from the global bifurcation at the BD point and connecting with the single-peak state in (a) are divided according to the number of peaks within each clump and are found in region II; region I contains equispaced structures organized in the foliated snaking structure generated in the RTB bifurcation at ${\mathrm{SN}}_b$ as described in Sec. 5b. For $\theta =2.5, {\rho }_{\mathrm{BD}}\approx 1.80278$ while ${\rho }_b\approx 1.80501$, so region I is very narrow.

Figure 13

(a) Bifurcation diagram showing the reconnection of the two-peak LP branch at $\theta =2.5$ (green curve) arising from homoclinic snaking at lower $\theta$ with the foliated snaking branches computed earlier in a global bifurcation at the BD transition (dashed line) shown in terms of the quantity $\mathrm{\Delta }·\left|\right|A-A_0^b{\left|\right|}^2$ to separate the different solution branches. The branch ${\mathrm{\Gamma }}_2$ of a periodic array of peaks, i.e., peaks separated by $L/2$, is shown in red. Bound states (BSs) with interpeak separation $\mathrm{\Delta }$ between the minimum separation of order $2\pi /\mathrm{Im}\left[\lambda \right]$ and $L/2$ are shown in black; these form by the pairwise locking of oscillatory tails in the region $\rho <{\rho }_{\mathrm{BD}}$. Solid (dashed) lines indicate stable (unstable) states. (b) The separation $\mathrm{\Delta }$ along the LS, ${\mathrm{\Gamma }}_2$, and BS branches. The vertical dashed line corresponds to the BD transition, while the dotted one to the ${\mathrm{SN}}_1^r$.

Figure 14

Destruction of the homoclinic snaking. (a) $\theta <2$: LPs (blue) undergo homoclinic snaking, and the different families of solutions arise from the HH bifurcation. (b) $\theta =2$: the pattern (red) involved in the formation of the LPs undergoes a global bifurcation at the QZ point, where its wavelength becomes infinite. At this point, homoclinic snaking is destroyed. (c) $\theta >2$: a global bifurcation occurs at the BD point. At this point, homoclinic snaking is destroyed. However, bright LS arise from the RTB bifurcation and undergo foliated snaking (green). The remnants of the homoclinic snaking branches (blue) reconnect with the branches forming the foliated snaking structure. The destruction of homoclinic snaking is indicated using $•$.