1. INTRODUCTION

Dissipative optical solitons are confined wave packets of light whose existence and stability require a continuous energy supply from an external source, which is dissipated in the medium around the solitons. This corresponds to an extension of the conventional soliton concept, implying that the single balance between nonlinearity and dispersion is replaced by a double balance: between nonlinearity and dispersion and also between gain and loss. Although the dissipative soliton concept evolved initially from phenomena observed or predicted in nonlinear optics, it has received wide recognition in other disciplines, such as biology and medicine [1,2].

A vast diversity of dissipative systems and nonlinear phenomena in physics, chemistry, and biology can be described by the complex Ginzburg–Landau equation (CGLE) [1–3]. In the field of nonlinear optics, the CGLE has been used to describe optical parametric oscillators [4], free-electron laser oscillators [5], spatial and temporal soliton lasers [6–10], and all-optical long-haul soliton transmission lines [11,12]. The active nonlinearity in the equation must be at least quintic in order to model the stable propagation of pulses [12]. Thus, the simplest model that we can use is the complex cubic-quintic CGLE. Depending upon the particular system or the phenomena that we want to describe, the model can be complemented with additional terms in the equation.

A large variety of soliton solutions can be obtained by numerically solving the cubic-quintic CGLE. In general, such solitons solutions belong to one of two classes: localized stationary solutions or localized pulsating solutions [1,12–14].

Among the localized pulsating solutions of the cubic-quintic CGLE, we may refer the plain pulsating, creeping, erupting, and chaotic solutions [14–22]. The regions of existence of pulsating solitons have been presented by Akhmediev et al. [14–22].

A creeping soliton is a pulsating localized solution that changes its shape periodically and shifts a finite distance in the transverse direction after each pulsation. It has a rectangular pulse with two fronts and one sink on the top [14], presenting a shape that resembles that of the composite soliton [8,14]. The two fronts pulsate back and forth relative to the sink asymmetrically at the two sides. Creeping solitons were first observed numerically in [14], and their existence was later numerically confirmed for various dissipative systems [14–24].

The first experimental observation of an erupting soliton was reported in 2002 by Cundiff et al. in a sapphire laser [25], whereas the plain pulsating soliton was observed for the first time in 2004 by Soto-Crespo et al. in a mode-locked fiber ring laser [26]. Nevertheless, the lack of a high-resolution real-time diagnostic method has precluded a detailed characterization of the pulsating behavior of such pulses. This limitation has been broken recently through the development of a novel powerful real-time spectra measurement technique called dispersive Fourier transform (DFT) [27]. Using this technique, the spectrum of the soliton could be mapped into a temporal waveform by using a dispersive element with enough group-velocity dispersion. Thus, the ultrafast spectral signals can be captured by a real-time oscilloscope with a high-speed photodetector. Actually, the DFT technique has enabled experimental observation in real time of transient dynamics of diverse pulsating solitons and other complex ultrafast nonlinear phenomena in fiber lasers [28–33].

Soliton self-organization and pulsation in a passively mode-locked fiber laser was observed experimentally in 2018 [29]. The soliton pulsation process showed a period corresponding to tens of the cavity round trip time. In another 2018 experiment, pulsating dissipative solitons in a mode-locked fiber laser at normal dispersion were observed for the first time using the DFT technique [30]. Also in 2018, the first experimental evidence of the pulsating soliton with chaotic behavior in an ultrafast fiber laser was reported [31].

In a 2019 experiment, three types of soliton pulsations were observed in an L-band normal-dispersion mode-locked fiber laser with the novel powerful real-time spectra measurement technique DFT [27,32]. They were classified as single-periodic pulsating soliton, double-periodic pulsating soliton, and soliton explosion. The evolution dynamics of creeping solitons in both the time and frequency domains in a mode-locked fiber laser were observed in 2020 by employing a Raman-assisted time-lens system operating in an asynchronous mode together with the DFT technique [33]. The periodical variation of pulse width, peak power, and motion range could be observed in real time, while the corresponding spectral evolution exhibited breathing dynamics.

If several solitons coexist in a laser cavity, they can form bound states, which are commonly referred to as soliton molecules (SMs). The dynamics of complex soliton interactions and observation of the motions within SM have been revealed by analyzing the real-time spectral dynamics using the DFT technique. Recently, Liu et al. reported the buildup dynamics of SMs in a mode-locked fiber laser [34]. Using the same technique, soliton splitting and movement have been recorded during the formation of harmonic mode locking [35].

Soliton interactions have been also widely investigated in different systems, namely in fiber lasers, as can be seen, for example, in [36] and references therein.

Theoretically, collisions of counter-propagating dissipative solitons can be investigated using two coupled one-dimensional cubic-quintic CGLEs. In this frame, phenomena like interpenetration and annihilation were for the first time reported in [37,38]. More recently, different outcomes resulting from the interaction of exploding dissipative solitons (DS) were reported [39,40].

In this work, we study the collision of two counter-propagating creeping solitons, whose behavior is described by the coupled complex cubic-quintic CGLEs. Five different outcomes are found as a result of such collision: periodic, fixed shape or quasi-fixed shape, interpenetration, plain pulses bound states with two frequencies, and a complex behavior. In Section 2, we present the two coupled CGLEs that govern the counter-propagation of pulses. In Section 3.A, we present the creeping soliton considered in the present work, and in Sections 3.B and 3.C, we show the results of collisions in the presence of positive nonlinear cross coupling $({c_r} \gt 0)$ and in the absence or negative nonlinear cross coupling $({{c_r} \le 0\;})$, respectively. The main results of this work are summarized in Section 4.

2. THEORY

In this paper, we consider a system of two coupled complex subcritical cubic-quintic Ginzburg–Landau equations to describe the propagation counter-propagating waves:

The numerical integration of Eqs. (1) and (2) was performed using a pseudo-spectral fourth-order Runge–Kutta method, together with the following initial conditions:

In general, we considered a time step $dt = 0.0018$, and as a grid spacing, we took $dx = 0.08$. The number of points has been changed from 1024 to 8192, in order to guarantee a higher spectral resolution, and confirm our results.

3. NUMERICAL RESULTS

A. Creeping Solitons

Figure 1(a) illustrates the behavior of a creeping soliton considering the following parameter values: $\mu = - 0.1, {\beta _r} = 0.8451, {\beta _i} = 1, {\gamma _r} = - 0.11, {\gamma _i} = - 0.08,\;{D_r} = 0.08,\;{D_i} = 0.5,{c_r},{c_i} = 0$. These values were selected considering the period-doubling bifurcation diagram presented in Fig. 14 of [21]. We have considered as an initial condition that $A({x,0}) = {\rm sech}(x),\;B({x,0}) = 0$, and ${x_{0}} = {0}$.

 

Fig. 1. (a) Space-time plot of a creeping soliton for ${300} \lt t \lt {600}$ and (b) pulse energy against the peak amplitude for $\mu = - 0.1, {\beta _r} = 0.8451, {\beta _i} = 1, {\gamma _r} = - 0.11, {\gamma _i} = - 0.08, {D_r} = 0.08, {D_i} = 0.5, v = 0$, and ${c_r} = {c_i} = 0$, assuming an initial condition obtained from Eq. (2), with $B = {0}$ and ${x_{0\:}} = {0}$. In (b), the starting point [peak amplitude (${t} = {300}$), energy (${t} = {300}$)] is represented by the circle. The arrow indicates the energy versus peak amplitude evolution from that point.

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The pulse profile evolution is shown in Fig. 1(a) for $t$ between 300 and 600, after achieving a truly periodic behavior. Actually, this type of solution corresponds to a single periodic pulsating soliton. It has been found that, when the value of ${\beta _r}$ is increased enough, the period of the pulsating soliton doubles [15]. Figure 1(b) shows the pulse energy against its peak amplitude. The periodic behavior of the creeping soliton is well demonstrated by the closed single curve in this figure. In the following, we will analyze the collisions between creeping solitons similar to the that in Fig. 1, obtained for $t \gtrsim 300$, in order to guarantee that such pulses have achieved their truly periodic state.

B. Collisions for ${{c}_{r}} \gt 0$

No collisions were observed for positive values of ${c_r}$, namely for $0 \lt {c_r} \le 0.4$, and pulse velocities $0 \lt v\; \le 0.5$. In this case, we only observed modulation instability, as illustrated in Fig. 2. This is due to the fact that the nonlinear cross coupling parameter is positive, ${c_r} \gt 0$, and will have a destabilizing effect, as expected.

 

Fig. 2. Modulation instability resulting from the interaction between two creeping solitons for ${c_r} = 0.2$, $v\; = 0.1$, and ${x_0} = 48$. The other parameters have the same values as in Fig. 1.

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C. Collisions for ${{c}_{r}} \le 0$

Figure 3 represents the phase diagram ${c_r}$ versus velocity, $v$, for the different outputs resulting from collisions between two creeping solitons. The range of the parameter values considered is ${-}0.4\; \le {c_r} \le 0$ and $0 \le v \le 0.5$, with incremental steps of 0.1. We found five different outputs as a result of the collision between two single periodic creeping solitons in the considered parameter regime: periodic (P), fixed shape or quasi-fixed shape (FS), interpenetration (IT), plain pulses bound states with two frequencies (BS), and complex behavior (CX). If ${c_r}$ is not zero, the outcomes are strongly dependent on the optical path, before the collision occurs.

 

Fig. 3. Phase diagram of different outputs of collisions between two creeping solitons, plotted as ${c_r}$ versus $v$. Solid circles indicate a periodic behavior (P); open circles represent fixed shape or quasi-fixed shape solutions (FS); open squares indicate interpenetration (IT); open diamonds represent bound states of plain pulses (BS), and finally open stars (CX) correspond to a complex behavior.

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It can be seen that for ${c_r} = 0$ the collisions result always in periodic solutions (P). The same is also true for high velocities and low absolute values of ${c_r}$. As the absolute value of ${c_r}$ increases, the fixed shape or quasi-fixed (FS) shape scenario becomes more likely. On the other hand, a slight change of the path can result in interpenetration (IT) of the pulses. Formation of bound states (BS) of plain pulses, at two different frequencies, is also observed. Finally, for small values of the pulse velocity, complex behavior (CX) is also observed. In the following, we present some results to illustrate each of these scenarios.

Figure 4 shows the periodic solution (P) for two different situations: ${c_r} = 0$ and ${c_r} = - 0.4$.

 

Fig. 4. Illustration of the periodic behavior (P) for two different situations: (a)–(c) correspond to ${c_r} = 0$, whereas (d)–(f) correspond to ${c_r} = - 0.4$. (a) and (d) show the trajectories in the plane ($x$, $t$) for ${c_r} = 0,\;\;v = 0.5, x_0 = 130$, and for ${c_r} = - 0.4,\;v = 0.2, x_0 = 90$, respectively. (b) and (d) show the peak amplitude evolution of both solutions A and B. $|{\rm B}|$ has been shifted by ${-}{1}$ in order to be distinguished from $|{\rm A}|$. (c) and (f) show the final amplitudes of A and B pulses [(a) ${{t}_{\rm{final}}} = \;{600}$ and (b) ${{t}_{\rm{final}}}\; = {800}$]. For convenience, the position $x$ of each pulse has been moved.

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In the case ${c_r} = 0$, Fig. 4(a) shows that the trajectory and the periodic evolution of pulses A and B remain unchanged after collision. This behavior was observed for all the range of velocities considered. In particular, Fig. 4(b) shows that the peak amplitude evolution of both pulses A and B do not change after collision. From Fig. 4(c), it can be seen that the final pulse shapes of both pulses are equal and symmetric, being similar to wide composite pulses [1,8].

 

Fig. 5. Illustration of fixed shape or quasi-fixed shape outcomes resulting from the collision between two creeping solitons for $v = {0.3}$ and ${c_r} = - 0.3$, assuming two different starting points: ${{x}_0} = {138}$ for (a)–(c) and ${{ x}_0} = {134}$ for (d)–(f). (a) and (d) show the trajectories in the plane ($x, t$), whereas (b) and (d) show the peak amplitude evolution of both solutions A and B. $|{\rm B}|$ has been shifted by ${-}{1}$ in order to be distinguished from $|{\rm A}|$. (c) and (f) show the final amplitude profiles of A and B [(a) $t_{\rm final} = 800$ and (b) $t_{\rm final} = 800$]. For convenience, the position $x$ of each pulse has been adjusted.

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Fig. 6. Interpenetration of two creeping solitons, producing three different outcomes. (a) and (b) for $v = {0.3}$, ${c_r} = - 0.4$, and ${{x}_0} = {140}$. (c) and (d) for $v = {0.1}$, ${c_r} = - 0.3$, and ${{x}_0} = {46}$. (e) and (f) for $v = {0.2}$, ${c_r} = - 0.3$, and ${{x}_0} = {73}$ [(a) ${{t}_{\rm{final}}} = {1400}$, (c) ${{t}_{\rm{final}}} = {1500}$, and (e) ${{t}_{\rm{final}}} = {800}$].

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A different scenario is observed in the case ${c_r} = -0.4$. From Figs. 4(d) and 4(e), it can be seen that the trajectories of pulses A and B are distorted, whereas their peak amplitude changes after collision. Moreover, Fig. 4(f) shows that the final pulse shapes become asymmetric, with the solutions A and B being a mirror image of each other. It was observed that the trajectory distortion occurs for all negative values of parameter ${c_r}$, but it seems to depend also on $v$.

Figure 5 shows two different fixed shape outcomes, considering the same parameter values, namely ${c_r} = - 0.3$ and $v = {0.3}$. However, slightly different paths were introduced assuming two different starting points: (a)–(c) ${x_0} = {138}$ and (d)–(f) ${x_0} = {134}$.

From Figs. 5(a) and 5(d), it can be seen that the periodic behavior is lost after pulse collision. However, the velocity is not the same for both cases. Figures 5(b) and 5(e) show that the peak amplitude evolution is constant or nearly constant in the two cases. Figures 5(c) and 5(f) illustrate two different scenarios: two moving solitons are formed in the first case, whereas two composite solitons are observed in the second case [2]. We conclude that a slight change of the starting point can produce a huge difference in the final output. Eventually, this may be due to the difference between the energies and the phases of the two pulses at the collision point. The two moving solitons are asymmetric solutions, corresponding to a mirror image of each other. In contrast, the composite pulses are symmetric solutions with a similar temporal profile. A composite soliton is considered to be a bound state of a plain pulse and two fronts, while a moving soliton is considered as a composite pulse where one of the fronts is missing [2].

Figure 6 illustrates three different outcomes in the presence of interpenetration resulting from the collision of two creeping solitons. Figures 6(a) and 6(b) illustrate a never-ending interpenetration of two moving solitons, which are a mirror image of each other. Figures 6(c) and 6(d) show that fixed shape pulses can emerge after a relatively long time of interpenetration (${\approx} {500}$). Such fixed shape pulses correspond to symmetric composite solitons that move at almost their initial velocity. Finally, Figs. 6(e) and 6(f) illustrate a less frequent case, in which each creeping soliton originates two composite pulses, but not at the same time. Actually, Fig. 6(e) shows that, soon after the collision of pulses A and B, four different pulses are formed. Two of them follow a straight trajectory, while the other two pulses experience interpenetration. After some time, the interpenetration ends, and these two pulses move along a straight trajectory, parallel to that of the first pair. The four pulses emerging from the collision are all composite pulses.

Figures 7(a) and 7(b) show bound state (BS) formation of two compound plain pulses. Actually, it is not frequent to find plain pulses as outputs from collisions of creeping solitons. However, we must retain that composite pulses are considered as bound states of a plain pulse and two fronts [1,8].

 

Fig. 7. Bound states of two compound plain pulses resulting from the collision of two creeping solitons, for $c_r = - 0.3, v = 0.3$, and $x_0 = 136$. (a) Trajectory in the plane ($x,\;t$); (b) amplitude profile of the two solutions A and B.

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Figure 8 shows complex behavior between the two pulses resulting from the collision between two creeping solitons for $v = {0.1}$, ${c_r} = - 0.4$, and ${{ x}_0} = {48}$. Interpenetration does not occur, in spite of the low value of $v$ and the high absolute value of ${c_r}$. However, the interaction between the two pulses is observed during a long time, during which they move at nearly zero velocity. Finally, after $t = {1000}$, the pulses move away with a velocity different from the initial one and no longer exhibit period one.

 

Fig. 8. Complex behavior resulting from the collision of two creeping solitons for ${v} = {0.1}$, ${c_r} = - 0.4$, and ${{x}_0} = {48}$.

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4. CONCLUSION

We have studied the collisions of two counter-propagating creeping solitons and found five classes of outcomes for the range of the velocities and the strength of the cubic cross coupling between them. We found periodic solutions, fixed shape or quasi-fixed shape solutions without interpenetration, solutions with interpenetration, bound state formation of two compound plain pulses, and solutions with complex behavior. On the other hand, we found that the final result is strongly dependent on the collision point. A small change of the optical path may result in very different scenarios. For higher values of the velocities and lower values of the cubic cross coupling parameter, periodic solutions seem to predominate. As the absolute value of the coupling parameter increases, fixed shape or quasi-fixed shape solutions tend to be dominant. However, when the magnitude of this parameter becomes greater than the velocity of the counter-propagating waves, other scenarios can occur.

To test experimentally the predictions of this paper, it appears natural to exploit the know-how already accumulated during the study of pulsating solitons in passively mode-locked fiber lasers. Although soliton pulsation and breathing dynamics have already been studied by utilizing the DFT technique, motion dynamics of soliton creeping have been hardly observed in experiment. Using a temporal magnifier (time-lens) system together with the DFT technique could help this observation.