1. Introduction

Dissipative solitons (DSs), commonly referred to the localized solutions in dissipative nonlinear system, result from the balance between nonlinearity and dispersion, gain and loss [1]. The concept of DSs has been exploited widely to describe nonlinear dynamics of far-from-equilibrium systems in different fields, including hydrodynamics, plasma physics, chemistry, biology and optics [2–4]. Because of its flexible configurations, the passively mode-locked fiber lasers (PMLFLs) are considered as an ideal platform to investigate diversified dynamic processes of the DSs [5]. Some real-time measurement techniques, such as the dispersive Fourier transform (DFT) [6] and time-lens [7], have provided the ability to measure the shot-to-shot spectra and intensity envelope with subpicosecond temporal resolution. More recently, the combination of these two techniques promises to obtain the full-field characterization of transient DSs dynamics [8]. Therefore, many known DSs phenomena, such as soliton molecule [9, 10], soliton explosion [11], rouge wave [12] and noise-like pulses [13], have been revisited experimentally to give an insight into these dynamic processes. At the same time, more numerical studies corresponding to these experimental results are also expected to further reveal the underlying physical mechanisms behind them.

Pulsating solitons, usually corresponding to periodic attractors in dissipative systems, are usually regarded as the intermediate regime between stable solitons and chaos. Pulsating soliton solutions with various evolutionary processes, such as plain, erupting, creeping, and the so-called “strong” pulsation, have been numerically demonstrated based on the complex Ginzburg-Landau equation (CQGLE) [14–18]. The dynamic processes of soliton pulsation, usually accompanied by periodic changes in the energy, are therefore affected by the gain saturation effect in the cavities of the PMLFLs. However, the gain dynamic effects have not been taken into account in these studies. Moreover, recent experiments have revealed more internal evolutionary processes in soliton pulsation through the DFT technique, such as vibration-like and phase drifting in the soliton molecule [9] and the appearance of Kelly sidebands periodically [19]. The Kelly sidebands in the spectra usually correspond to the dispersive waves that are generally considered to be emitted by solitons due to the perturbations of lumped components in the cavity [20]. More recently, Kelly sidebands have also been found during the buildup of soliton molecules [10]. However, the role of the dispersive waves in these phenomena has yet to be further studied. Therefore, it is of great significance to further explore numerically the details of the dynamics of the DSs pulsation.

In this study, we investigate the DSs pulsation in the PMLFLs based on CQGLE with the items of gain saturation. The evolution and transition of DSs pulsation are illustrated by bandwidth-energy trajectories and the cross-correlation frequency-resolved optical gating (XFROG) traces. With the increase of linear gain coefficient and gain saturation intensity, a single soliton becomes unstable, and then the single-periodic pulsation, soliton molecule, snaking pulsation, and double-periodic pulsation appear sequentially. We show that, even without considering the effect of the lumped components, the dispersive wave will still appear periodically, accompanied by the pulsating process. In our case, the generation of the dispersive wave is explained by two mechanisms. In single-periodic pulsation, the dispersive wave is generated from the soliton radiating excess energy, while in the snaking and double-periodic pulsation, it is generated from the periodic collision of the solitons. The dynamic processes of the double-periodic pulsation and soliton molecule formation from pulsation are illustrated by the XFROG traces. We show that, under certain saturation intensity, an additional soliton can be generated by the disturbance caused by the dispersion wave and forms a soliton molecule. These phenomena are qualitatively consistent with the recent experiments of soliton pulsations and soliton molecules [9, 10, 19]. Moreover, if the additional soliton entangles with the original one by periodically collisions, the snaking pulsation and double-periodic pulsation will occur. They have complex or even chaotic dynamics and may be intrinsically related with the unstable condensed states of the DSs in the PMLFLs.

2. Model

The propagation of dissipative solitons in the PMLFLs is modeled by CQGLE with the terms of the gain saturation, namely CQGL-type equation [21, 22]:

In dissipative systems, the attractor is a fixed localized solution. The evolution processes of the DSs will converge into the same attractor after several round trips even if the initial condition has a small difference. We use the single stationary solution of CQGL-type equation as the initial condition to study the influence of the gain saturation term. In this case, I_s = ∞ and Γ = 0, the gain dynamic effect is not considered.

We present the evolution trajectories of the DSs in a two-dimensional bandwidth-energy space (σ_F, Q) to describe the feature of these dynamics. The total energy in the calculated temporal window [t1, t2] is defined as $Q={\displaystyle {\int }_{t1}^{t2}|E{|}^{2}dt}$, while the spectral width, ${\sigma }_F=\sqrt{\sqrt{〈f^2〉-{〈f〉}^2}}$, where f is the normalized frequency and 〈f ⁿ〉 stands for $\frac{1}{Q}{\displaystyle {\int }_{-\infty }^{\infty }{f}^n|E{|}^{2}df}$. In order to apply to the more complicated evolution process, such as multi-soliton and chaos, we avoid using the position, amplitude and phase of a single soliton to represent the evolution trajectory. To better understand correlated temporal and spectral features during the pulsation and transition between different states of the DSs, we also calculate the XFROG traces S by:

3. Simulation results

At first, the effect of the gain saturation term to soliton dynamics is investigated with the initial condition of a single soliton as mentioned above. A set of calculations, starting from the initial condition, varying g₀ and I_s and fixing the value of other parameters at D = 1, ε = 0.58, β = 0.025, µ = −0.12, υ = −0.1, r = 2, Γ = 0, are performed to investigate how the gain coefficient g₀ and saturation intensity I_s impact soliton dynamics. After thousands of round-trips evolution, the stable states are obtained which are summarized in Fig. 1. As we can see, it is divided into six regions on a (g₀, I_s) plane, corresponding to different dynamic regimes labeled as single steady soliton, single-periodic pulsating soliton, soliton molecule, snaking pulsating soliton, double-periodic pulsating soliton and complex region.

 

Fig. 1 A map of soliton behaviors of CQGL-type equation solutions in the parameter plane (g₀, I_s).

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3.1. Single steady soliton

In Fig. 1, region I corresponds to the single steady solution of the CQGL-type equation. The evolution of temporal intensity profile at g₀ = 2.4, I_s = 0.32, point 1 in Fig. 1, is demonstrated in Fig. 2(a). As we can see, the feature of the soliton remains stable. Note that, there is slightly oscillation at the beginning (not visible in the figure) which gradually disappears after propagating for several round-trips. Figure 2(b) shows the spectrum at z = 3000, which clearly shows the Kelly sidebands. Hence, the soliton is steady in this region after a short transitional process.

 

Fig. 2 (a) Evolution of temporal intensity profile at point 1 in Fig. 1 with g₀ = 2.4 and I_s = 0.32. (b) Spectrum at z = 3000.

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3.2. Single-periodic pulsating soliton

When I_s is slightly increased, the single stationary soliton becomes unstable and single-periodic pulsation occurs. The single-periodic pulsating soliton region is sandwiched between the single steady soliton region and the soliton molecule region, as shown in Fig. 1. Figure 3(a) is the evolution of the temporal intensity profile at g₀ = 2.4, I_s = 0.324 in this strip region, point 2 in Fig. 1. After undergoing a transition process at the beginning, the soliton exhibits a stable periodic change corresponding to closed stripes in the picture. Because the period is relatively small compared to the entire evolutionary process, the details are not observable and are shown following. Figure 3(b) is the evolution trajectories of the soliton in (σ_F, Q) space and consists of two diagrams. The upper one corresponds to the transition process (from 1 to 1000 round-trips), in which the trajectory spirals outward and converges to a periodic attractor eventually. The lower one corresponds to the stably pulsating behavior (from 1000 to 6000 round-trips), where the final trajectory constitutes a closed circle, indicating that the evolution repeats itself and the oscillation has a single period. In each period, the energy Q varies from 13.7 to 6.66. Figures 3(c) and 3(d) are the enlarged diagrams of the temporal profile and spectrum, from 3000 < z < 3030, respectively. These diagrams show that the temporal width and spectrum are periodically broadening during the pulsation, corresponding to the circle trajectory in Fig. 3(b). Note that the spectral sidebands can be observed periodically, which means that the soliton periodically radiates energy through the dispersive wave.

 

Fig. 3 (a) Evolution of temporal intensity profile at the point 2 in Fig. 1 with g₀ = 2.4 and I_s = 0.324. (b) The evolution trajectories in (σ_F, Q) space. The upper one is for 1 < z < 1000, the lower one is for 1000 < z < 6000. (c) The enlarged diagrams of the time profiles from 3000 < z < 3030. (d) The spectrum corresponds to (c).

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3.3. Formation of soliton molecule

In Fig. 1, the region III is the soliton molecule region in which the single-periodic pulsating soliton is unstable and splits in two or three new stationary solitons with fixed spacing and phase. Figure 4(a) illustrates the evolution of the temporal intensity profile at g₀ = 2.4, I_s = 0.33, point 3 in Fig. 1. As shown in this figure, soliton molecule with two solitons, spontaneously appears after the unstable pulsating solitons vanishes at z = 2340. The enlarged figure of the soliton molecule generation can be seen in Fig. 4(b). The center of the pulsating soliton moves back and forth and when moving to the left maximum position, a new soliton on the left side of the original soliton is generated. After several round-trips, these two solitons become stable and the detailed process is shown in Fig. 4(e). The spectrum of the soliton molecule has an interference fringe with a profile similar to that of a single soliton. Figure 4(c) is the spectrum of evolution corresponding to Fig. 4(b). Figure 4(d) shows the evolution trajectory in (σ_F, Q) space when 2200 < z < 6000. In this figure, the blue curve deviates from the circular trajectory of a single periodic pulsation when 2200 < z < 2370, while the red one represents the processes of converging to the fixed point (0.38, 14.94) when 2370 < z < 6000, corresponding to the formation of the soliton molecule. Figure 4(e) shows the XFROG traces of the transition process from the pulsation to soliton molecule (see Visualization 1). During the pulsation, the soliton radiates a dispersive wave that corresponds to the Kelly sidebands in spectrum, as shown in the subplots in Fig. 4(e), Figs. 4(e1) and 4(e2), corresponding to two different states within one period. Because the figure is static, the radiation process of the dispersion wave cannot be clearly seen in these figures while it can be clearly seen in the video. Then, the pulsation becomes unstable, and when it moves to the left maximum position, a new soliton appears. It is interesting that the new soliton is generated from the substrate of the dispersive wave rather than splitting of the original one, which can be seen in Figs. 4(e3) and 4(e4). After that, the two solitons attract and collide with each other, and then merge into one soliton, as shown in Fig. 4(e5). When the collision occurs, they dissipate more energy through dispersive wave. After evolving into several round-trips around a period of the original pulsation, the merged soliton divides to two solitons as before. However, these two solitons are quickly separated, until repulsion and attraction are balanced and a stable soliton molecule is formed, as shown in Fig. 4(e6). Note that, a soliton molecule with three solitons can also be formed from the pulsation of a single soliton, indicating that this region has multistability.

 

Fig. 4 (a) Evolution of the time profiles of the soliton at the point 3 in Fig. 1 with g_o = 2.4 and I_s = 0.33. (b) The enlarged evolution of (a) for 2100 < z < 2500. (c) Spectrum corresponding to (b). (d) The evolution trajectory in (σ_F − Q) space when 2200 < z < 6000. (e) The XFROG traces during the formation of the soliton molecule. (e1), (e2), (e3), (e4), (e5) and (e6) are at z = 1499.2, 1501.2, 2322.1, 2327.4, 2330.8 and 2500, respectively (see Visualization 1).

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3.4. Double-periodic pulsating soliton

The region V in Fig. 1 is referred to as the double-periodic pulsating soliton. Figures 5(a) and 5(b) are the evolution of temporal intensity profile and spectrum at g₀ = 2.4 and I_s = 0.528, point 8 in Fig. 1, for 3000 < z < 3400. These figures display that the pulse repeats itself after two periods with the periods being around 5 and 6 in z direction, respectively. The evolution trajectory in (σ_F, Q) space is shown in Fig. 5(c), which consists of two tangent circular tracks. It indicates that the pulsating soliton returns to its original state after two different cycles. Figure 5(d) is the XFROG traces in a pulsating period (see Visualization 2). The upper figures, Figs. 5(d1) and 5(d2), are in the first circle while the lower figures, Figs. 5(d3) and 5(d4), are in the second circle within one period. From the XFROG traces, we find that the double-periodic pulsating soliton is actually composed of two twining solitons that attract each other and collide periodically. At each collision, the spectrum broadens significantly and the Kelly sidebands appear. At the same time, the energy of the solitons is emitted as dispersive waves (clearly seen in Visualization 2). Thus, compared with Ref. [15] and Ref. [18], the pulsating solutions in our model, single-periodic and double-periodic pulsating soliton, are periodically radiates energy through the dispersive wave to maintain the pulsating process. Moreover, the interaction of two twining solitons constitute the two different periods of double-periodic pulsating soliton.

 

Fig. 5 (a) and (b) are the evolution of temporal intensity profile and spectrum of 3000 < z < 3400 of point 8 in Fig. 1 with g₀ = 2.4 and I_s = 0.528, respectively. (c) Evolution trajectory in (σ_F, Q) space for 140 < z < 6000. (d) XFROG traces in a pulsating period. (d1), (d2), (d3) and (d4) are at z = 3005.6, 3008.4, 3011.1 and 3014.1, respectively (see Visualization 2).

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3.5. Snaking pulsating soliton

The region IV in Fig. 1 corresponds to an unstable double-periodic pulsating soliton, referred to as the snaking pulsating soliton, in which the pulsating solitons change their position along a zig-zag track in time domain. Snaking pulsation is a combination of the shape and positional oscillations of the solitons, which exhibits complex and variable dynamic processes. Figure 6(a) shows the temporal evolution of a snaking pulsating soliton at g₀ = 2.3 and I_s = 0.676, point 4 in Fig. 1, which is derived from a double-periodic pulsating at 2700 round-trip and then monotonously moves to the left of the time window as a whole with a constant drift velocity. When we zoom in the figure, as shown in Figs. 6(c) and 6(e), the zig-zag motion in time and double-periodic pulsating in shape are simultaneously found. Figure 6(b) shows the evolution trajectories of the double-periodic pulsating (2300<z<2700) and snaking pulsating soliton (2700<z<6000) in (σ_F, Q) space. Comparing these two figures, we can clearly see that the trajectory of the snaking pulsation is derived from that of the double-periodic pulsation and the solitons returns to their initial state after 7 double-periodic pulsations, which means that the period of the zig-zag movement is exactly 7 times the pulsating period. Figures 6(d) and 6(f) are the spectra of the snaking pulsation corresponding to Figs. 6(c) and 6(e), respectively. A periodic change in the center frequency can be clearly observed. Since the center frequency corresponds to the group velocity, the frequency deviation causes the motion of the soliton relative to the frame of reference in time, and attributes it to the zig-zag track. It is noted that, limited by the spectral gain bandwidth β, the normalized center spectrum of solitons can only fluctuate around zero in our model. Not all snake-like movements of the soliton are regular and synchronized with the double-periodic pulsating. Figures 7(a)–7(d) demonstrate a snaking pulsating soliton, point 5 in Fig. 1, with an irregular center position and a quasi-periodic zig-zag motion. Figure 7(c), the enlarged figure of Fig. 7(a), shows that the position of the zig-zag motion in time domain is different on each side, corresponding to the asymmetrical change of the center frequency. Therefore, we can determine that the evolution trajectory in (σ_F, Q) space is chaotic. Figures 8(a)–8(d), points 6 and 7 in Fig. 1, show two kinds of snaking pulsation with integral zero-drift velocity in time. Although their evolutions are very similar in the temporal intensity profile, as shown in Figs. 8(a) and 8(c), a significant difference can be found in their trajectories in (σ_F, Q) space. In Fig. 8(b), the trajectory is chaotic, which means that the double-periodic pulsation and zig-zag motion are out of tune. Whereas, in Fig. 8(d), the two kinds of period motion are synchronous and the solitons can return to their initial position after 10 periods of the doubling periodic pulsation.

 

Fig. 6 (a) Evolution of the temporal intensity profile for the snaking pulsating soliton, point 4 in Fig. 1 with g₀ = 2.3 and I_s = 0.676. (b) Evolution trajectory in (σ_F, Q) space. The upper figure is for 2300 < z < 2700, the lower figure is for 2700 < z < 6000. (c) and (e) are the enlarged evolution of temporal intensity profiles for 3000 < z < 4000 and 3000 < z < 3200, respectively. (d) and (f) are the enlarged evolution of spectrum for 3000 < z < 4000 and 3000 < z < 3200, respectively.

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Fig. 7 (a) Evolution of temporal intensity profile for the snaking pulsating soliton, point 5 in Fig. 1 with g₀ = 2.4 and I_s = 0.506. (b) Evolution trajectory in (σ_F, Q) space for 1300 < z < 6000. (c) and (d) are enlarged evolution of temporal intensity profiles and spectrum for 3000 < z < 4000.

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Fig. 8 (a) Evolution of the temporal intensity profile for the snaking pulsating soliton, point 6 in Fig. 1 with g₀ = 2.5 and I_s = 0.412. (b) Evolution trajectory in (σ_F, Q) space. (c) Evolution of the temporal intensity profile for snaking pulsating soliton, point 7, in Fig. 1 with g₀ = 2.6 and I_s = 0.34. (d) Evolution trajectory in (σ_F, Q) space.

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Region V is a complex space, in which more solitons can exist simultaneously due to higher gain parameters. These solitons can combine the various states as we described above. For example, a stable soliton molecule and a double-periodic pulsating soliton can coexist in the cavity.

4. Discussion

Multistability is a common phenomenon in dissipative systems, which means that different initial conditions can converge to different attractors. In the previous simulation, we all use a single soliton as the initial condition. To further understand the characteristics of the pulsating dynamics, we also use a soliton molecule with two or three solitons as the initial condition of the simulation, with fixed g₀ = 2.4 and varying I_s. In Fig. 9, the solid curves shows the energy of the stable soliton or soliton molecule, while the gray regions, corresponding to that in Fig. 1, illustrate the variation range of the soliton energy during the pulsation. As shown in the figure, stable double and triple soliton molecules can exist in a wide range of I_s. The single-period pulsation occurs when a single soliton becomes unstable, which is a typical Hopf bifurcation. However, the snaking pulsation and double-periodic pulsation are significant different from the single-periodic pulsation. Although they are made up of two solitons, when they appear, the soliton molecule is still stable under the same parameter space. If we compare the generation processes of the soliton molecule and double-periodic pulsation from a single soliton, we can find they are very similar in the initial stage of the transition from the single-periodic pulsation. For both cases, two solitons are formed from the single-periodic pulsating soliton and then collide with the other one. After the collision, however, they experience different dynamic processes. If the two solitons are quickly separated, a stable soliton molecule can be formed. Otherwise, the two solitons are entangled and periodically collided with each other to form the snaking pulsation or double-periodic pulsation. This difference could be understood as follow. As the gain saturation intensity I_s increases, the repulsive force between the solitons decreases. Therefore, in the short distance, the repulsive force between the solitons is not sufficient to separate the solitons to form a stable molecule.

 

Fig. 9 DSs multistability diagram with varying I_s and initial condition. The rest of parameters are fixed: D = 1, ε = 0.58, β = 0.025, µ = −0.12, υ = −0.1, r = 2, g₀ = 2.4 and Γ = 0.

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Dispersive waves, corresponding to the Kelly sidebands in spectra, is a very common phenomenon in passive mode-locked fiber lasers [23]. In previous studies, dispersive waves were generally thought to be caused by the perturbation of the soliton by the lumped components in the cavity [20]. However, although we do not consider the effects of the lumped components in our model, the Kelly sidebands and dispersive wave can still be observed periodically, accompanied by the pulsating process. This means that the dispersion wave may have some new generation mechanisms. In our case, the dispersive wave generation could be explained by two mechanisms. In the single-periodic pulsating process, the steady soliton loses its stability due to the increase in the gain saturation intensity. Therefore, the solitons need to dissipate excess energy through the dispersive wave to maintain them around a particular attractor in a dissipative system. Whereas, for the snaking pulsation and double-periodic pulsation, a strong dispersive wave is emitted when two solitons collide with each other. In this case, the dynamic instability induced by the collision could be attributed for the generation of dispersive wave.

The snaking pulsating soliton and double-periodic pulsation are composed of two entangled solitons that attract each other and periodically collide. The phenomena are not only related to the generation of the soliton molecule, but also possibly to some unstable states with Kelly sidebands, such as the condensed phase in soliton rain. However, to the best of our knowledge, these phenomena have not been observed experimentally. One of the possible reasons could be that they are very easily confused with unstable traditional solitons. We hope that the entangled solitons can be observed by real-time measurement methods in the future.

We also investigate the influence of the fast response of the gain term in this system. A single-periodic pulsating soliton, point 2 in Fig. 1, is used to study how the gain dynamics modify the behavior of the solitons. Except Γ, other parameters of equation retain the data used to study the influence of the gain saturation term. Figures 10(a) and 10(b) show the time profiles, both from 1 < z < 3000, of single-period pulsating solitons with Γ = 0.0001 and Γ = 0.001, respectively. It is shown that Γ of a small value forces the solitons to move in horizontal when propagating. When Γ increases, the instability of the soliton increases and the dynamics may change.

 

Fig. 10 (a) and (b) are the spatial evolution of single-period soliton, point 2 in Fig. 1 with Γ = 0.0001 and Γ = 0.001.

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5. Conclusion

In summary, we numerically observe the single-periodic pulsation, soliton molecule, snaking pulsation, and double-periodic pulsation, induced by gain characteristics of dissipative system. The generation mechanisms of the dispersive wave are investigated in the model without lumped components. We find that the dispersive wave plays the key role in the processes of soliton pulsating and the soliton molecule formation. These results would be in aid in understanding the phenomena of soliton pulsation and the formation of soliton molecule in recent real-time spectra experiments. The snaking pulsation and double-periodic pulsation correspond to the unstable and stable entangled soliton states, in which more solitons could be generated. This work will stimulate further experimental research on the internal dynamics of the complex soliton bunches in dissipative systems, such as soliton liquid, the condensed phase in soliton rain and noise-like pulse.