Interaction of dissipative solitons stabilized by nonlinear gradient terms

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.

Figure 1

Phase diagram of possible outcomes of collisions of stationary NLGS DSs for $R_r=0.1$ plotted as $c_r$ versus velocity $v$. Filled black circles ($•$) indicate stationary bound states, open black circles ($\circ$) oscillatory bound states, open blue triangles meandering oscillatory bound states ($▵$), red solid circles partial annihilation ($•$), green solid squares large-amplitude oscillatory bounds states ($\blacksquare$), blue diamonds annihilation ($⧫$), and black open squares ($\square$) interpenetration. The parameters are as follows: $\mu =-0.012, {\beta }_r=0.3, {\beta }_i=1.0, D_r=0.6$, and $D_i=0.5$.

Figure 2

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the temporal generation of a stationary bound state indicated by the black solid circles in Fig. 1 for $c_r=-0.3$ and $v=0.1$. The timescale shown is $T=360$ and the box size $L=62.5$.

Figure 3

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the temporal generation of an oscillatory bound state indicated by the black open circles in Fig. 1 for $c_r=-0.3$ and $v=0.4$. The timescale shown is $T=110$ and the box size $L=62.5$.

Figure 4

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the time evolution of a meandering oscillatory bound state indicated by the open blue triangles in Fig. 1 for $c_r=-0.5$ and $v=0.7$. The timescale shown is $T=360$ and the box size $L=62.5$.

Figure 5

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the time evolution of a large-amplitude oscillatory bound state indicated by the green solid squares in Fig. 1 for $c_r=-0.5$ and $v=1.0$. The timescale shown is $T=360$ and the box size $L=62.5$. We note that the large-amplitude oscillations are combined with some degree of meandering on timescales large compared to the frequency of the oscillations.

Figure 6

Four snapshots of $\left|A\right|$ as a function of space $x$ during one period $T$ and plotted $T/4$ apart for the large-amplitude oscillatory bound state; $c_r=-0.5$ and $v=1.0$.

Figure 7

Two time series of the quantity $I_A\left(t\right)\equiv \int \left|A(x,t)\right|dx$ are shown for the meandering oscillatory bound state for $c_r=-0.5$ and $v=0.7$ from $t=30$ to $t=600$.

Figure 8

The Fourier spectra $S\left(\omega \right)$ for meandering oscillatory bound states ($c_r=-0.5$ and $v=0.7$) (top) and for large-amplitude oscillatory bound states ($c_r=-0.5$ and $v=1.0$) (bottom) are plotted for comparison. The range covered for $\omega$ is $0\le \omega \le 5$. We note the change of scale on the ordinate. While the spectrum for the meandering oscillatory bound state is clearly showing signatures of chaotic behavior, the spectrum for the large-amplitude oscillatory bound state shows predominantly periodic behavior with the fundamental and the first few harmonics.

Figure 9

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the partial annihilation indicated by the red solid circles in Fig. 1 for $c_r=-0.3$ and $v=1.0$. The $x-t$ plot includes both the initial approach and the temporal behavior after the partial annihilation is completed. The timescale shown is $T=110$ and the box size $L=62.5$.

Figure 10

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the annihilation indicated by the blue solid diamonds in Fig. 1 for $c_r=-0.3$ and $v=1.5$. The $x-t$ plot includes both the initial approach and the temporal behavior after the annihilation is completed. The timescale shown is $T=40$ and the box size $L=62.5$.

Figure 11

The $x-t$ plot of Max $\left(\right|A|,|B\left|\right)$ showing the process of interpenetration indicated by the open black squares in Fig. 1 for $c_r=-0.1$ and $v=1.6$. The $x-t$ plot includes both the initial approach and the temporal behavior after the annihilation is completed. The timescale shown is $T=30$ and the box size $L=62.5$.

Figure 12

Transition from a stationary bound state to oscillatory bound states. $c_r=-0.5$. (a) Fourier spectrum giving rise to $\omega =5.76$ for $v=0.11$; (b) amplitude $R$ as a function of $v$ demonstrating that it is a Hopf bifurcation $R\sim {(v-v_c)}^{\gamma }$ with $v_c=0.104$ and an exponent $\gamma =0.50$; (c) $\omega$ is plotted as a function of $v$ and fitted to a straight line as predicted by Eq. (8).

Figure 13

In this plot we show the first period doubling in the regime of oscillatory bound states for $c_r=-0.5$. The Fourier spectra are for three values of the velocity $v$: (a) $v=0.22$, the state before the first period doubling; (b) $v=0.23$, at the transition to the first period doubling, with the new amplitude still fairly small; and (c) $v=0.24$ first period doubling completed with fairly large amplitude for the doubled period.

Figure 14

In this plot we show the evolution of the Fourier transforms for larger values of the velocity, $v$, in the regime of oscillatory bound states for $c_r=-0.5$. (a) $v=0.30$, a new period doubling occurs; (b) $v=0.315$, new frequencies are starting as visible from the Fourier spectrum; (c) $v=0.40$, fully developed chaotic state.

Figure 15

Determination of period and amplitude for the large-amplitude oscillatory bound state for $c_r=-0.5$ and $v=1.0$: $x-t$ plot showing the amplitude A of the wave traveling to the right for a shorter period of time and $L=62.5$. For the amplitude we obtain $\mathrm{\Delta }=4.09$ and for the period $T$ we get $T=11.1$.

Figure 16

Determination of amplitude $\mathrm{\Delta }$ and period $T$ for the large-amplitude oscillatory bound state for $c_r=-0.5$ as a function of velocity $v$: (a) amplitude $\mathrm{\Delta }$ plotted as red solid circles ($•$) as function of $v$; (b) period $T$ plotted as filled black circles ($•$) as a function of $v$. In both cases we observe stronger than linear growth with growth rates that closely parallel each other. At $v=1.26$ annihilation of the two DSs replaces large-amplitude oscillatory bound states.

Figure 17

The period $T$ indicated by filled black circles ($•$) and the rescaled amplitude, $\mathrm{\Delta }$ (rescaled by a factor $8/3$) shown as red solid circles ($•$) from Fig. 16 are plotted as a function of velocity $v$. We observe that that the curves superpose almost perfectly.

Figure 18

The Fourier spectra for meandering oscillatory bound states for two values of the velocity $v$ and $c_r=-0.5$. $v=0.60$ (top) and $v=0.70$ (bottom). The range covered for $\omega$ is $0\le \omega \le 10$. While both Fourier spectra clearly showing signatures of chaotic behavior, the width of the Fourier spectrum as well as the background noise level are reduced as the velocity increases from $v=0.60$ (top) to $v=0.70$ (bottom).

Figure 19

The Fourier spectra for meandering oscillatory bound states for two values of the velocity $v$ and $c_r=-0.5$. $v=0.80$ (top) and $v=0.90$ (bottom). The range covered for $\omega$ is $0\le \omega \le 10$. We note the change in scale on the ordinate compared to Fig. 18. We note that the width of the Fourier spectrum and the background noise level are reduced as the velocity increases.