Relative stability of multipeak localized patterns of cavity solitons

We study the relative stability of different one-dimensional (1D) and two-dimensional (2D) clusters of closely packed localized peaks of the Swift-Hohenberg equation. In the 1D case, we demonstrate numerically the existence of a spatial Maxwell transition point where all clusters involving up to 15 peaks are equally stable. Above (below) this point, clusters become more (less) stable when their number of peaks increases. In the 2D case, since clusters involving more than two peaks may exhibit distinct spatial arrangements, this point splits into a set of Maxwell transition points.

Figure 1

(a) Bifurcation diagram of Eq. (1) showing two interweaved snaking curves, one of them corresponding to odd numbers of peaks and the other to even numbers of peaks in the localized pattern. Solid (dotted) curves correspond to stable (unstable) localized solutions. Parameters are $C=-{10}^{-4}$ and $\Gamma =0.06$. (b) Localized patterns with 1, 3 and 5 peaks. (c) Localized patterns with 2, 4, and 6 peaks. Panels (b) and (c) are obtained for the same values of parameters and $Y=-0.002$.

Figure 2

Dimensionless Lyapunov functional $\mathcal{L}$ as a function of the control parameter $Y$. Solid (dotted) curves correspond to stable (unstable) localized solutions. $M$ is the Maxwell point. Parameters are the same as those in Fig. 1a. The branches $A1$, $B1$, and $C1$ correspond to the LPs shown in Fig. 1b. The branches $A2$, $B2$, and $C2$ correspond to the LPs shown in Fig. 1c.

Figure 3

Same as Fig. 2a, but calculated for clusters made up of up to 15 peaks. Enlarged view of the vicinity of the Maxwell point $M$. The points calculated numerically are shown by black dots.

Figure 4

Stable clusters of 2D hexagonal close-packed peaks of Eq. (1) on the ($x$,$y$) plane (a) and dependencies of the Lyapunov functional on the injection parameter $Y$ for these clusters (b). $M1$, $M2$, and $M3$ are the intersection points. Parameters are $C=-0.05$ and $\Gamma =3/8$. Maxima are plain black.