Anomalous Diffusion of Dissipative Solitons in the Cubic-Quintic Complex Ginzburg-Landau Equation in Two Spatial Dimensions

We demonstrate the occurrence of anomalous diffusion of dissipative solitons in a “simple” and deterministic prototype model: the cubic-quintic complex Ginzburg-Landau equation in two spatial dimensions. The main features of their dynamics, induced by symmetric-asymmetric explosions, can be modeled by a subdiffusive continuous-time random walk, while in the case dominated by only asymmetric explosions, it becomes characterized by normal diffusion.

Figure 1

(a)–(g) 2D snapshots of the random time evolution of the modulus $|A(x,y;t)|$ of an asymmetric exploding soliton. $\mu =-0.1$, ${\beta }_r=1$, ${\beta }_i=0.8$, ${\gamma }_r=-0.1$, ${\gamma }_i=-0.6$, $D_r=0.125$, $D_i=0.5$. (a) $t=0$, (b) $t=200$, (c) $t=400$, (d) $t=600$, (e) $t=800$, (f) $t=1000$, (g) $t=1200$. (h) 3D snapshot of the DS between explosions. Maximum of $|A(x,y,t)|$ is around 2. In the plane $(x,y)$ the DS is localized inside a circle of radius 10.

Figure 2

Comparison of two different diffusive behaviors of the $x$ coordinate of the center of mass of the soliton. For each value of $\mu$, the results of ten independent realizations with slightly different initial conditions were superimposed. (a) $\mu =-0.30$, (b) $\mu =-0.35$.

Figure 3

Ensemble-averaged MSD defined by Eq. (3), evaluated for the same two cases as in Fig. 2. $N=128$ slightly different initial conditions were used in each case. (a) $\mu =-0.30$, (b) $\mu =-0.35$.

Figure 4

Time-averaged MSD defined by Eq. (4), evaluated for the same two cases as in Fig. 2. $N=128$ slightly different initial conditions were used in each case. (a) $\mu =-0.30$, (b) $\mu =-0.35$.

Figure 5

Statistics of the normalized time-averaged MSD defined in Eq. (5) (with $\mathrm{\Delta }={10}^4$, $T={10}^5$) extracted from the soliton behavior. (a) For $\mu =-0.30$, the histogram of $\xi$ is compared with $\varphi (\xi )$ associated with Brownian motion. (b) For $\mu =-0.35$, the histogram of $\xi$ (using $\mathrm{\Delta }=10$, $T={10}^5$) is compared with the distribution defined in Eq. (6) associated with a subdiffusive 2D CTRW, using $\alpha =0.75$ extracted from the ensemble-averaged MSD [see Fig. 3].