Stability analysis of numerically exact time-periodic breathers in the Lugiato-Lefever equation: Discrete vs continuum

We describe a framework for numerical calculation of time-periodically oscillating (breather) solutions to the discretized Lugiato-Lefever equation (LLE), as well as their linear stability as obtained from numerical Floquet analysis. Compared to earlier approaches, our work allows for the following conclusions: (i) The complete families of solutions are obtained also in regimes of instability; (ii) analysis of Floquet spectra and the corresponding eigenvectors show clearly the nature of the various Hopf and period-doubling bifurcations; (iii) properties of breather solutions to the continuous LLE are connected to corresponding oscillating solutions of the discrete LLE, which is of interest in its own right modeling coupled nonlinear cavities. In particular, we show that the oscillating discrete cavity solitons found in earlier work can be viewed as lattice versions of the continuous LLE breathers, as there is a smooth continuation in parameter space connecting them. Moreover, we confirm the existence of stable breathers at large detunings that was recently observed experimentally and describe their appearance from bifurcations.

Figure 1

Real parts of Floquet eigenvalues versus frequency detuning $\mathrm{\Delta }$ when $C=75$, $E_0=4$, and $N=121$ sites.

Figure 2

(a) Real parts of Floquet eigenvalues versus coupling constant $C$ when $\mathrm{\Delta }=-7.0$, $E_0=4$, and $N=121$ sites. The figure is cut at the lower boundary, and the two strongly unstable eigenmodes for $22\lesssim C\lesssim 49$ have extrema with eigenvalues of approximately $-4$ and $-7$, respectively. (b) Time period $T$ versus $C$. The lower branch corresponds to the family continued from larger $C$ while the upper branches correspond to the bifurcating breather families, which are always unstable. Also shown are snapshots for $C=40$ of (c) breather intensity $|u_n{|}^2$, and two corresponding eigenmodes of the linearized problem with eigenvalues on the negative real axis: (d) antisymmetric unstable and (e) symmetric stable (which will however destabilize for slightly smaller $C$).

Figure 3

(a) Real parts of Floquet eigenvalues versus driving strength $E_0$ when $\mathrm{\Delta }=-7.0$, $C=99$, and $N=121$ sites. (b) Snapshots for $E_0=4.01$ (solid) and $E_0=6$ (dashed) of breather intensity $|u_n{|}^2$. (c) Unstable eigenmode for $E_0=4.5$.

Figure 4

(a) Real parts of Floquet eigenvalues versus coupling constant $C$ when $\mathrm{\Delta }=-3.0$, $E_0=2.0$, and $N=121$ sites. (b) Snapshot for $C=10$ of (unstable) breather intensity $|u_n{|}^2$. (c) Unstable eigenmode for $C=10$.

Figure 5

(a) Real parts of Floquet eigenvalues versus coupling constant $C$ when $\mathrm{\Delta }=-4.0$, $E_0=3.0$, and $N=121$ sites. (b) Snapshots of breather intensity $|u_n{|}^2$ for $C=2.5$ (solid line, strongly discrete) and $C=50$ (dashed line, continuum-like). (c) Time period $T$ versus $C$.

Figure 6

(a) Real parts of Floquet eigenvalues versus frequency detuning $\mathrm{\Delta }$ when $E_0=45$, $C=700$, and $N=121$ sites. (b) Time period $T$ versus $\mathrm{\Delta }$. The purple line corresponds to the continuation in (a), the blue line to the bifurcating solution at $\mathrm{\Delta }\approx -96.93$, and the green line to the bifurcating solution at $\mathrm{\Delta }\approx -54.75$. The inset in (b) shows magnification around the bifurcation point $\mathrm{\Delta }\approx -54.75$, where yellow dots indicate twice the period of the fundamental stable breather existing at these parameter values. (The intersections with the dotted line do not correspond to bifurcations.)

Figure 7

Initial dynamics of breathers at various positions in Fig. 6. (a) Stable breather on the purple branch at $\mathrm{\Delta }=-85$. (b) Unstable breather on the same branch at $\mathrm{\Delta }=-55.8$. Also shown are breathers for the same parameter values but corresponding to (c) the stable breather with half the period [yellow dots in the inset in Fig. 6] and (d) the unstable breather on the blue branch. All parameter values are the same as in Fig. 6.