Bound Pulse Trains in Arrays of Coupled Spatially Extended Dynamical Systems

We study the dynamics of an array of nearest-neighbor coupled spatially distributed systems each generating a periodic sequence of short pulses. We demonstrate that, unlike a solitary system generating a train of equidistant pulses, an array of such systems can produce a sequence of clusters of closely packed pulses, with the distance between individual pulses depending on the coupling phase. This regime associated with the formation of locally coupled pulse trains bounded due to a balance of attraction and repulsion between them is different from the pulse bound states reported earlier in different laser, plasma, chemical, and biological systems. We propose a simplified analytical description of the observed phenomenon, which is in good agreement with the results of direct numerical simulations of a model system describing an array of coupled mode-locked lasers.

Figure 1

(a) Schematic representation of a ring array of nearest-neighbor coupled mode-locked lasers. (b) Laser intensities in the bound pulse train regime in a ring of four lasers calculated for $\eta =0.5$ and $\phi =3.0$. Different colors correspond to different lasers. Other parameters: $\gamma =33.3$, $\kappa =0.1$, ${\alpha }_g=2.0$, ${\alpha }_q=3.0$, $\vartheta =0$, $G_0=2.0$, $Q_0=4.0$, ${\gamma }_g=0.0133$, ${\gamma }_q=1$, $s=25$, and $\tau =1.875$ (similarly to Ref. [36]).

Figure 2

(a) Bifurcation diagram in coordinates $(\phi ,\eta )$ for the synchronized solutions in the ring array with $N=4$. Left- and right-inclined hatching indicates the stability regions for the in-phase ($l=0$) and antiphase ($l=2$) synchronized solutions. Red lines $P$ correspond to pitchfork bifurcations of the in-phase synchronized solution. (b) Bifurcation diagram for the bound state $\mathbf{B}\mathbf{1}$ (cf. Fig. 4) in the plane $(\phi ,\eta )$. The same as in panel (a), in addition, the light gray area shows the stability domain of the bound state. The red line $P$ corresponds to a supercritical pitchfork bifurcation of the in-phase synchronized solution leading to the bound pulse train regime, the blue line $F$ corresponds to a fold bifurcation where stable and unstable branches of bound pulse train solutions merge and disappear, and the dashed black line $T$ shows the first torus bifurcation of the pulse bound state which leads to a modulation in a form of slight change of the pulse shapes from one pulse period to another.

Figure 3

(a) Space-time diagram of the bound pulse train regime in a four-laser array in coordinates $(T,\sigma )$, where $\sigma =t\text{mod}{\tau }_0$ is the original fast time and $T=t/{\tau }_0$ the slow time (${\tau }_0=1.9054$). Brighter colors indicate higher values of the sum of the laser intensities $\sum _{i=1}^4|A_i{|}^2$. (b) Space-time diagram for the pulse train bound state regime for two lasers $({\tau }_0=1.9043)$. Coupling parameters are $\eta =0.5$ and $\phi =3.0$.

Figure 4

Stable equilibria and their basins of attraction on the phase plane of the reduced system (7) for coupling phase $\phi =3.0$. $\mathbf{C}\mathbf{1}$ corresponds to the stable antiphase synchronized solution. Equilibria $\mathbf{B}\mathbf{1}$, $\mathbf{B}\mathbf{2}$, and $\mathbf{B}\mathbf{3}$ correspond to bound states with increasing time separation $\mathrm{\Theta }$, which have different phase shifts $\mathrm{\Phi }$ between pulse intensity maxima. Inset: An example of the intertwining basins of attraction of five stable bound states in the vicinity of a unstable spiral source for Eq. (7) for $\phi =3.99$.