Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

We study the stability and bifurcation structure of spatially extended patterns arising in nonlinear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number $k_c$ arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number $2k_c$. This next branch also connects to a branch of patterns with double wave number, this time $4k_c$, and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $\theta =2$ from below the critical wave number $k_c$ approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.

Figure 1

Stable HSS (black solid line) is destabilized at the modulational instability MI. Close to MI (i) the unstable HSS evolves to the pattern branch ${\mathrm{P}}_1$ (red) consisting of stationary patterns with wave number $k_1=0.706849\approx k_c=0.707107$. Further away from MI (ii) the unstable HSS evolves into a different pattern branch ${\mathrm{P}}_2$ (green), associated with patterns with wave number $k_2=0.824564\approx k_u=0.836642$. Stable (unstable) solutions are denoted by solid (dashed) lines. Here $\theta =1.5$ and $L=160$.

Figure 2

Left: marginal instability curves for (a) $\theta =1.1$, (b) $\theta =1.5$, (c) $\theta =1.8$, and (d) $\theta =2.0$. Right: the HSS solutions corresponding to the same values of $\theta$. Solid (dashed) lines represent stable (unstable) HSSs with respect to perturbations of the form (6). The locations ${\mathrm{I}}_k^{\pm }$ corresponding to instabilities with wave number $k$ are indicated using solid circles. The dashed line inside the marginal instability curves in the left panels represents the most unstable mode $k=k_u$.

Figure 3

(a) Instability lines ${\mathrm{I}}_{k_c}^{\pm }$ and the location of the saddle-node bifurcations of the HSSs in the parameter space $(\theta ,I_0)$. (b) Same as (a) but in the parameter space $(\theta ,\rho )$. (c) Zoom of (b) showing the main regions with distinct bifurcation behavior (see text). The labels ${\mathrm{X}}_1$ and ${\mathrm{X}}_2$ indicate codimension-two points. In both (a) and (c) the gray area represents region V where the HSS is triple valued.

Figure 4

Comparison between the asymptotic solution (14) (blue solid line) and the corresponding numerically exact solutions (red diamonds) obtained using a Newton-Raphson solver (see Sec. 4) for (a),(b) a supercritical pattern at $\theta =1.1$ and (c),(d) a subcritical pattern at $\theta =1.5$. In (a) and (c) the real part $U$ is shown, while (b) and (d) show the imaginary part $V$. In both cases the numerical and analytical curves are almost indistinguishable. In both cases $L=n2\pi /k_c$, with $n=16$, and $|\rho -{\rho }_c|={10}^{-5}$.

Figure 5

Bifurcation diagrams for patterns with wave numbers $k_c, 2k_c$, and $4k_c$ for $\theta =1.5$. Solution profiles along the different branches obtained from numerical continuation are shown in panels (i)–(xii).

Figure 6

Phase diagram in the $(\theta ,\rho )$ parameter space showing the main bifurcations of the HSS and pattern states. In (a),(b) the regions of existence of ${\mathrm{P}}_{k_c}$ and ${\mathrm{P}}_{2k_c}$ are shaded in light gray. In (a) the region of bistability between $A_0^b$ and ${\mathrm{P}}_{k_c}$ is indicated in dark gray. For clarity the bifurcation lines EC and FWH corresponding to the Eckhaus and Hopf bifurcations are omitted in these two panels. Panel (c) shows the full phase diagram, including the EC and FWH lines, with the region of stability of both patterns shown in light blue. Panel (d) shows a close-up view of (c) around the cusp bifurcation C. The symbol $•$ represents the codimension-two points ${\mathrm{X}}_1$, C, ${\mathrm{D}}_1, {\mathrm{D}}_2$, and ${\mathrm{D}}_3, {\mathrm{Z}}_1$, and ${\mathrm{Z}}_2$.

Figure 7

Bifurcation diagrams corresponding to (a) $\theta =1.1$, (b) $\theta =1.3$, (c) $\theta =1.4$, (d) $\theta =1.5$, (e) $\theta =1.6$, and (f) $\theta =1.8$. Red lines correspond to ${\mathrm{P}}_{k_c}$ and the blue lines to ${\mathrm{P}}_{2k_c}$. Panels (a) and (b) show the situation before and after the codimension-two point ${\mathrm{X}}_1$. Panels (b) and (c) show the transition from supercritical to subcritical bifurcation of pattern ${\mathrm{P}}_{k_c}$ via a degenerate HH at $\theta =41/30$. For $\theta =1.5$ [panel (d)] ${\mathrm{P}}_{2k_c}$ bifurcates supercritically from HSS at ${\mathrm{I}}_{2k_c}^{-}$. In contrast, for $\theta =1.6$ [panel (e)] ${\mathrm{P}}_{2k_c}$ emerges subcritically. Solid (dashed) lines indicate stable (unstable) branches.

Figure 8

Phase diagram in $(\theta ,\rho )$ parameter space showing an enlargement of the diagrams shown in Fig. 6 focusing on the main stability regions of ${\mathrm{P}}_{k_c}$ labeled ${\mathrm{II}}_{\mathrm{A},\mathrm{B}}$ and ${\mathrm{P}}_{2k_c}$ labeled ${\mathrm{III}}_{\mathrm{A},...,\mathrm{C}}$. The dashed line at $\theta =1.5$ refers to the slice of this diagram shown in Fig. 9. The small black dots along this line correspond to points where the stability analysis of the patterns shown in Sec. 5 was performed. In light blue we show the regions where ${\mathrm{P}}_{k_c}$ and ${\mathrm{P}}_{2k_c}$ are stable.

Figure 9

Bifurcation diagram for $\theta =1.5$. The pattern branch ${\mathrm{P}}_{k_c}$ (red) bifurcates subcritically from HSS at ${\mathrm{I}}_{k_c}^{-}$, while the branch ${\mathrm{P}}_{2k_c}$ (blue) bifurcates supercritically at ${\mathrm{I}}_{2k_c}^{-}$. Labels (a)–(c) correspond to the unstable patterns with 32 rolls initially that evolve in time to patterns with different numbers of rolls depending on the value of $\rho$ and lying on new branches of periodic states (gray) labeled by ${\mathrm{P}}_n$, where $n$ is the new roll number. The points where linear stability analysis has been carried out are indicated using the symbol $•$.

Figure 10

Eigenspectrum in the vicinity of the EC instability of the ${\mathrm{P}}_{2k_c}$ branch when $\theta =1.5$, showing $\mathrm{Re}\left[{\lambda }_1\left(q\right)\right]$ for different values of $\rho$: (a) $\rho =1.57$, (b) $\rho ={\rho }_{{\mathrm{EC}}_2}=1.58$, and (c) $\rho =1.59$.

Figure 11

$\mathrm{Re}\left[{\lambda }_1\left(q\right)\right]$ at $\theta =1.5$ in the region of Eckhaus instability and the associated temporal evolution of an unstable initial pattern to patterns of different wavelengths. These new states are shown in gray in Fig. 9: an unstable pattern with initially 32 rolls evolves to ${\mathrm{P}}_{25}$ in panel (c) for $\rho =1.4$, to ${\mathrm{P}}_{24}$ in panel (b) for $\rho =1.3$, and to ${\mathrm{P}}_{22}$ in panel (a) for $\rho =1.2$. The left panels show the unstable modes $0<q<q^{*}$, while the right panels describe the resulting evolution in space-time plots.

Figure 12

Eigenspectrum of ${\mathrm{P}}_{2k_c}$ in the vicinity of the FW instability when $\theta =1.5$, showing the first two branches $\mathrm{Re}\left[{\lambda }_1\left(q\right)\right]$ and $\mathrm{Re}\left[{\lambda }_2\left(q\right)\right]$ for different values of $\rho$: (a) $\rho =1.175$, (b) $\rho ={\rho }_{\mathrm{FW}}\approx 1.177$, and (c) $\rho =1.179$.

Figure 13

Hopf bifurcation of ${\mathrm{P}}_{2k_c}$ at $\theta =1.5$ showing (left panels) $\mathrm{Re}\left[\lambda \right(q\left)\right]$ for different values of $\rho$: (a) $\rho =1.82$, (b) $\rho ={\rho }_{\mathrm{FWH}}=1.87$, and (c) $\rho =1.92$. The right panels show the corresponding eigenspectrum at $q=k_c$, the onset wave number.

Figure 14

Time evolution of the oscillating patterns for $\theta =1.5$ and (a) $\rho =1.9$, (b) $\rho =2.1$, and (c) $\rho =2.3$.