Extreme soliton pulsations in dissipative systems

We have found a strongly pulsating regime of dissipative solitons in the laser model described by the complex cubic-quintic Ginzburg-Landau equation. The pulse energy within each period of pulsations may change more than two orders of magnitude. The soliton spectra in this regime also experience large variations. Period doubling phenomena and chaotic behaviors are observed in the boundaries of existence of these pulsating solutions.

Figure 1

(a) Periodic emergence of spikes by the pulsating soliton found for the following set of parameters of Eq. (1): $D=-1.0,\nu =0.1,\delta =-0.1,\beta =0.125,\epsilon =0.95$, and $\mu =-0.001$. (b) The evolution of the spectrum for the same simulation. The spectrum expands approximately eight times when the pulse reaches its highest amplitude. Another specific feature of the spectrum is the set of sidebands at fixed frequencies generated after the spike.

Figure 2

Pulse profile evolution of the same solution as in Fig. 1 during one period. The pulse experiences a sharp amplitude increase while keeping its width nearly constant. The spike then decays and two smaller pulses leave the central one symmetrically on each side. They gradually vanish. This can be seen more clearly in Fig. 1.

Figure 3

Pulse peak amplitude evolution for the same solution as in Fig. 1. The two vertical dotted lines are the boundaries of the $z$ interval of evolution in Fig. 2. The highest amplitude $\approx 12.5$ exceeds approximately five times the lowest amplitude that is $\approx 2.7$.

Figure 4

Pulse energy evolution for the same solution as in Fig. 1.

Figure 5

Bifurcation diagram representing the maximal $Q_M$ (red dashed line) and minimal $Q_m$ (blue dotted line) values of the energy of the periodic solution vs $D$. Other parameters, written in this figure, are the same as in Fig. 1. The ratio $Q_M/Q_m$ versus $D$ is shown by the solid green curve on the same dimensionless scale as $Q$. In the gray vertical areas, solutions vanish on propagation, no matter what kind of initial conditions are used.

Figure 6

Magnification of a small part of the bifurcation diagram of Fig. 5 near the first gray stripe. Only maxima of the energy $Q_M$ are plotted in this diagram. Period doubling bifurcation of solitons occurs at $D\approx -0.503$. It follows by period quadrupling at $D\approx -0.50245$. Chaotic evolution can be observed in a narrow region of $D$ around $\approx -0.5022$.

Figure 7

Central amplitude evolution of the soliton at the point $D=-0.5032$ of the bifurcation diagram of Fig. 6, i.e., to the left of the point of period doubling bifurcation. This is an example of the most extreme oscillations: the maximum of the peak amplitude exceeds almost 20 times the minimum amplitude. The period is also larger than in the case of Fig. 3 while the spatial width of the spike is significantly smaller than the period.

Figure 8

The energy evolution of the soliton for the same data as in Fig. 7. This is an example of the most extreme pulsations: the maximum energy exceeds the minimum energy more than 115 times. The average energy is shown by the blue dashed line above the gray area in this plot.

Figure 9

Extreme changes of (a) the shape and (b) the spectrum that the pulse experiences in a single period for $D=-0.5302$. The two profiles in (a) are shown for the pulse when it has the maximum amplitude (blue dashed curve) and minimal amplitude (red area). The two spectra in (b) correspond to the same profiles. The spectrum of the pulse widens significantly and splits into several sidebands with each one stronger than the spectrum shown in red.

Figure 10

Bifurcation diagram vs $D$ when parameter $\mu =-0.003$. The solid black line represents the energy $Q$ of stationary solutions while the red dashed line (blue dotted line) represents the maxima (minima) of the energy of pulsating solitons.

Figure 11

Bifurcation diagram vs $D$. Parameter $\mu =-0.002$. The solid black line represents the energy $Q$ of stationary solutions while the red dashed line (blue dotted line) represents the maxima (minima) of energy of the pulsating solitons. This diagram has a discontinuity: when moving from the left (increasing $D)$ the solution vanishes at $D=-0.226$. Another branch of pulsating solutions with smaller amplitudes starts at slightly larger $D$. When reaching the discontinuity point from the right, the solution jumps directly to the left-hand side branch. This transition is indicated by the black arrow.

Figure 12

Bifurcation diagram vs $\mu$. The soliton is stationary with fixed energy $Q$ on the left-hand side of $\mu \approx -0.002$. The soliton becomes periodic to the right-hand side of this value. The minimal $Q_m$ (dotted blue curve) and maximal $Q_M$ (dashed red curve) values of $Q$ quickly diverge with the increase of $\mu$. The dashed black vertical line corresponds to the example in Fig. 1. The upper limit of $Q_M$ at $\mu \approx -0.0004$ seems to be unlimited.

Figure 13

Evolution of (a) the profile, (b) the spectrum, and (c) the energy of the pulsating soliton very close to the bifurcation point $\mu =-0.002$ in Fig. 12. Energy in this case oscillates almost harmonically.

Figure 14

Evolution of (a) the profile, (b) the spectrum, and (c) the energy of the pulsating soliton at the point $\mu =-0.0005$ of the bifurcation diagram in Fig. 12. Pulsations are highly nonlinear.

Figure 15

Areas of existence of (red) stationary and (blue) pulsating solitons on the $\left(D,\mu \right)$ plane. Solutions disappear in the black spots. The thick pink dots correspond to the solutions given in the figures above labeled with the same number. The vertical and horizontal dashed lines are chosen to construct the bifurcation diagrams shown above. The solutions cannot be resolved numerically for values of $\mu$ that are too small as in the white stripe on top of this plot.

Figure 16

(a) Soliton energy evolution in the area of chaotic dynamics with $\mu =-0.0007$ and $D=-0.637$. The false color plot in (b) shows the soliton splitting and merging at around the point $z\approx 110$ in (a).

Figure 17

The XFROG diagram of the pulse when its maximal amplitude is about to be reached. The spectrum of the pulse widens significantly and splits into several highly energetic sidebands. The sidebands on each side are specifically related to one of the trails of the pulse in the time domain.