Dissipative systems are a class of open, non-equilibrium physical systems that maintain certain dynamic behaviors by exchanging energy or matter with their environment under conditions far from thermodynamic equilibrium [1]. Mode-locked fiber lasers are typical dissipative systems, where intricate balances between dispersion and nonlinearity, as well as gain and loss, lead to complex dissipative soliton dynamics such as soliton pulsation, soliton explosions, soliton molecules, and rogue waves [2–6]. This makes them an excellent test bed for studying the dynamical behaviors of dissipative systems.

Traditional mode-locked lasers based on single-mode fibers can be regarded as one-dimensional dissipative systems. By considering properties such as light’s polarization, spatial distribution, and wavelength, more complex and higher-dimensional dissipative phenomena can be realized, including various forms of vector solitons [7–9] and spatiotemporal mode-locking and related soliton dynamics [10–12]. In recent years, the control of multi-wavelength dimensions in fiber laser cavities has attracted significant attention. The solitons with different central frequencies can be mutually captured through the cross-phase modulation (XPM) effect, thus forming a composite structure that overlaps in the temporal domain and propagates with the same group velocity (GV) [13–16]. This, in turn, leads to a time-frequency coupling dynamical process, such as specific temporal and spectral vibration characteristics [17], dissipative Talbot solitons [18], stripe solitons [19], heteronuclear multicolor soliton compounds [20], and dichromatic breather molecules [21]. These studies have effectively suppressed the GV difference and gain competition between pulses of different wavelengths by employing various methods, such as spectral pulse shapers [18,20,22,23], multi-channel gain techniques [17,21], and polarization-maintaining fiber [19]. These approaches have enabled various types of synchronized soliton compounds in fiber resonators, providing a novel solution for multi-wavelength specialized pulse sources [24] and spectral analysis [25]. However, no studies have yet demonstrated the asynchronous control of multiple soliton compounds with different wavelengths within a mode-locked laser cavity.

Collisions between solitons are crucial interactions in dissipative systems. In addition to triggering phenomena such as the collision dynamics of vector solitons [26,27], the evolution of noise into a mode-locked state at a new wavelength induced by dual-wavelength soliton collisions [28], the shedding of dispersive waves [29], and Hopf-type bifurcation reversible leaps during dual-wavelength soliton molecule collisions [30], they are often accompanied by soliton annihilation and regeneration, along with dynamic processes such as the formation and dissociation of soliton molecules [31,32], elastic and inelastic recombination [33,34], chaotic explosions [35], soliton oscillations [36], and soliton state transitions in three-wavelength pulse collisions [37]. Although collisions between solitons and soliton molecules [34], as well as intra-pulse interactions within soliton molecules [32] and even between dual- and tri-color solitons [35–37], have been observed in passively mode-locked lasers, the collision dynamics and energy conversion mechanisms between multi-wavelength heteronuclear soliton compounds remain unknown. Investigating the collision mechanisms of these multi-wavelength heteronuclear soliton compounds not only opens up new avenues for exploring soliton dynamics but also contributes to a deeper understanding of the interactions between chemical molecules.

In this study, we construct a single-mode passively mode-locked fiber laser (PMLFL) that supports two types of synchronized dichromatic soliton compounds and their collisions. A section of highly nonlinear fiber (HNLF) with a zero-dispersion wavelength near 1533 nm is inserted into the cavity to facilitate the formation of multi-wavelength compounds. We record the instantaneous dynamical processes both spectrally and temporally in real time using time-stretch dispersion Fourier transform (TS-DFT) technology [38]. The XPM effect leads to the formation of a robust fast-GV compound (FGC), consisting of a partially coherent dissipative soliton bunch (PCDSB) and dispersion waves (DWs), while a conventional soliton (CS) and a narrow spectral pulse (NSP) form a slow-GV compound (SGC). Multiple SGCs can further interact to form an SGC loosely bound complex. These two types of compounds with different GVs continuously collide and exchange energy through the four-wave mixing (FWM) effect in the HNLF, promoting the annihilation, survival, and regeneration of the SGC complex. Importantly, experimental results highlight the indispensable role of the FWM effect induced by HNLF and the capture processes resulting from the XPM effect in the regeneration of the SGC complex. Furthermore, this work reveals the general relationship between regeneration behavior, FWM, and mutual self-trapping effects in the dynamics of multi-wavelength soliton compound collisions. It also provides new insights into the dynamics of complex dissipative soliton systems and offers potential methods for information encoding and processing based on mode-locked lasers.

The experimental setup for the multi-wavelength passively mode-locked fiber laser (PMLFL) and real-time detection system is illustrated in Fig. 1. The synchronized multichromatic PMLFL includes a 2.8 m erbium-doped fiber (EDF) as the gain medium and a 0.7 m HNLF (YOFC NL-1550-Zero, nonlinear coefficient $\gamma =10.5{\mathrm{W}}^{-1}/\mathrm{km}$) with a zero-dispersion wavelength at approximately 1533 nm, which enhances the intracavity nonlinear effects. It also includes 8.8 m of single-mode fiber, comprising both fiber components and pigtails. The group velocity dispersion (GVD) values of three types of the fibers are 40, $-0.4$, and $-23{\mathrm{ps}}^2/\mathrm{km}$ at 1550 nm, respectively. The repetition rate of the mode-locked fiber laser cavity is 16.7 MHz. The gain fiber is pumped by a 976 nm pump, and the pump and signal light are coupled using a 980/1550 nm wavelength division multiplexer (WDM). A polarization-dependent isolator (PD-ISO) ensures unidirectional laser oscillation in the cavity. Two polarization controllers (PCs) are used to fine-tune the intracavity birefringence. Passive mode-locking is achieved through the nonlinear polarization rotation (NPR) effect. A 3:7 1550 nm optical coupler (OC) is employed to output part of the mode-locked pulse energy. The evolution of the output pulses is recorded in real-time with high temporal resolution using high-speed oscilloscopes (Tektronix DPO75902SX) with a bandwidth of 59 GHz and a sampling rate of 200 GS/s, achieving a time resolution of 5 ps and a spectral resolution of 0.086 nm along with a TS-DFT system. The laser output is divided into three branches using two single-mode optical couplers (OCs). One branch directs the pulse signal to a 45 GHz bandwidth photodetector (PD1, InGaAs PIN detector, HSPD4040) for real-time intensity detection, while another pulse signal passes through a 1.5 km dispersion compensating fiber with total dispersion of $-197.01\mathrm{ps}/\mathrm{nm}$ at 1545 nm [DCF-G.652, C/250, $-131.34\mathrm{ps}/(\mathrm{nm}·\mathrm{km})$ at 1545 nm] and is synchronously measured by another photodetector (PD2, FINISAR, XPDV3120R) for real-time stretched spectral analysis. To facilitate shot-to-shot spectral observation and analysis, a fiber optical attenuator (FOA) is placed before the DCF to appropriately adjust the incident pulse power. It is important to note that although there is a certain time delay between the two branches, this does not affect our analysis of the simultaneously measured pulse spectra and temporal intensity. Furthermore, another branch directs the pulse to an optical spectrum analyzer (OSA, Yokogawa, 70D) for synchronous monitoring of the pulse’s average spectrum, with a spectral resolution set at 0.02 nm. The black and red dashed boxed regions in Fig. 1 also illustrate the conceptual diagram of intracavity pulse states and soliton dynamic evolution inside the cavity.

The resonant cavity of the laser exhibits overall anomalous dispersion. By adjusting the polarization controller (PC) and pump power, mode-locking can be achieved using the NPR effect. Additionally, variations in intracavity birefringence and polarization-dependent loss affect the spectral filtering within the EDF gain spectrum, leading to different transmission peaks at various wavelengths [39]. This wavelength-dependent dual balance between gain and loss, as well as nonlinearity and dispersion, promotes the emergence of multichromatic pulse mode-locked states. The inserted zero-dispersion HNLF facilitates the generation, mutual interaction, and gain equalization of multichromatic pulses through the FWM effect, enabling these pulses to stably coexist. By setting the pump power to 160 mW, we obtained a typical multichromatic mode-locked spectral structure on the optical spectrum analyzer (OSA), which can be divided into three regions, as shown in Fig. 2. The green dashed line at $\sim 1533\mathrm{nm}$ corresponds to the zero-dispersion wavelength of the HNLF. Given the high thermal dissipation efficiency of the fiber and the predominant dissipation of generated heat through thermal exchange with the surrounding environment, the thermal effect of the pump power within the cavity on the HNLF is negligible. To the left of the green dashed line, a broad-spectrum pulse with a central wavelength at $\sim 1531.3\mathrm{nm}$ is observed. At longer wavelengths, a CS spectrum with a central wavelength at $\sim 1555.2\mathrm{nm}$ appears, marked by distinct Kelly sidebands [40]. The spectrum in the middle region is divided into two parts by a red dashed line. As shown in the enlarged view in the top right corner, the spectrum on the left exhibits a single-peak structure with a central wavelength at $\sim 1543.2\mathrm{nm}$, while the right side presents a multi-peak structure ranging from $\sim 1543.6\mathrm{nm}$ to $\sim 1545.4\mathrm{nm}$, attributed to instability. Based on the spectral results, we calculate that the frequencies ${\omega }_1(\sim 1531.3\mathrm{nm})$, ${\omega }_2(\sim 1543.15\mathrm{nm})$, ${\omega }_3(\sim 1555.2\mathrm{nm})$ in the HNLF satisfy the frequency matching condition for degenerate FWM [41], which is expressed as ${\omega }_1+{\omega }_3=2{\omega }_2=2.443\mathrm{THz}$. Since both the CS and broad-spectrum pulse exhibit wide gain spectral ranges, the wavelength ranges of the single-peak and multi-peak structures satisfy the frequency matching condition for degenerate FWM. In terms of phase matching, a linear effect is described by $\mathrm{\Delta }{k}_L\approx {\mathrm{\Omega }}_L^2{\beta }_2+{\mathrm{\Omega }}_L^4{\beta }_4/12$, where ${\mathrm{\Omega }}_L={\omega }_1-{\omega }_2$. The condition must satisfy $0>\mathrm{\Delta }{k}_L>-\mathrm{\Delta }{k}_{\mathrm{NL}}$ to achieve FWM gain, where $\mathrm{\Delta }{k}_{\mathrm{NL}}=2\gamma P_0$ represents the nonlinear effect, with $P_0=10\mathrm{W}$ being the peak power within the nonlinear fiber. For our analysis, we use second-order dispersion ${\beta }_2=-0.257{\mathrm{ps}}^2/\mathrm{km}$ and fourth-order dispersion ${\beta }_4=-0.000147{\mathrm{ps}}^4/\mathrm{km}$ at approximately 1544 nm for FWM in the HNLF, resulting in $0>\mathrm{\Delta }{k}_L=-23.1{\mathrm{km}}^{-1}>-\mathrm{\Delta }{k}_{\mathrm{NL}}=-210{\mathrm{km}}^{-1}$. Furthermore, the 0.7 m HNLF is significantly shorter than the FWM coherence length $L_{\mathrm{coh}}=2\pi /|\mathrm{\Delta }{k}_L|=272.1\mathrm{m}$. These results confirm that, upon propagation in the HNLF of the laser cavity, the pulse structures at four wavelengths form a gain equilibrium through energy transfer facilitated by the FWM effect, maintaining stable coexistence within the resonant cavity.

To gain deeper insights into the multichromatic pulse mode-locked state (see Fig. 2) and the dynamic evolution of intracavity interactions, we employed a high-speed oscilloscope to directly measure the time-domain pulse evolution over 10,000 round trips, as shown in Fig. 3(a). At the same time, we provide the original time-domain pulses for six round trips, as shown in Fig. 3(b). Additionally, we utilized the TS-DFT technique to capture the shot-to-shot spectral evolution over the same 10,000 round trips, as presented in Fig. 3(c). To facilitate the observation of the CS in the solid-line circular region of Fig. 3(c), an enlarged view of this region is shown on the right. The evolution diagram reveals two distinct pulse structures with different GVs, which inevitably lead to collisions. Solitons exhibiting anomalous dispersion in the cavity show slower GVs at longer wavelengths. We used the GV of the CS pulse as a reference, adjusting its trajectory perpendicular to the $x$-axis of the evolution diagram. This resulted in the faster broad-spectrum pulse displaying a positive slope. The $x$-axes in Figs. 3(a) and 3(c) are limited to a 26 ns range, which captures the primary dynamics over the 10,000 round trips. The TS-DFT evolution diagram in Fig. 3(c) shows that both the fast-GV and slow-GV structures split into two spectral components. In the fast-GV structure, the right side corresponds to the short-wavelength broad-spectrum pulse component from the average spectrum shown in Fig. 2, while the left side corresponds to the single-peak structure in the middle spectral region. In the slow-GV structure, the stretched spectral component on the left corresponds to the long-wavelength CS in the average spectrum, while the right side corresponds to the multi-peak region of the middle spectrum. To avoid saturation, attenuation was applied during measurement, causing the CS component to appear less distinct in Fig. 3. However, the CS component becomes more visible when the attenuation rate is reduced.

Following the application of the TS-DFT technique, both the fast-GV and slow-GV pulse structures were observed to separate into distinct compounds, as shown in Fig. 3(c), a characteristic not observed in direct measurements. The green curve on the right side of Fig. 3(c) illustrates the evolution of the total intracavity pulse energy. Notable energy fluctuations occur between the 3750th and 5250th round trips, during which collisions take place. The data before the TS-DFT process [see Figs. 3(a) and 3(b)] show that the main pulse in the FGC is accompanied or overlapped by smaller pulses, while several closely spaced pulses in the SGC overlap synchronously with the base pulses. The temporal shape of the individual SGC closely resembles that of a single soliton. Figure 3(c) shows a more noticeable separation between the two peaks in the FGC, due to the normal dispersion of the DCF used for TS-DFT. Simultaneously, the base pulses of SGC show significant separation from several closely adjacent pulses, supporting the observed bimodal spectral feature of the SGC in Fig. 3(c). The region inside the black dashed line in Fig. 1 shows a schematic diagram of the two types of soliton compounds with different velocities in the cavity, before and after stretching. These structures overlap in the time domain and separate in the frequency domain. The SGC, containing CS, and the FGC, containing a broad-spectrum pulse, are both formed by the overlap of components at two different wavelengths. The components of these two asynchronous compounds correspond to different wavelengths, as shown by the spectrum from OSA.

To further analyze the composition of the FGC, we extracted the temporal evolution and the TS-DFT evolution diagram for this structure over 1500 cavity cycles prior to the collision, as shown in Figs. 4(a) and 4(b). This compound has a cavity period of 59.7795 ns, corresponding to a group velocity of 0.205756 m/ns, with the two components propagating in parallel and exhibiting synchronized propagation properties. For clarity, we redrew the two-dimensional (2D) evolution diagram so that its trajectory aligns with the $y$-axis. The right side of Fig. 4(a) displays a strong broadband pulse with weak interference, corresponding to the broad-spectrum pulse centered at $\sim 1531.3\mathrm{nm}$ in Fig. 2. The time-domain spectra in Fig. 4(b), which corresponds to Fig. 4(a), reveal the presence of a partially coherent dissipative soliton bunch with a width of several hundred picoseconds and noticeable jitter characteristics. Thus, the weak coherence observed in the DFT spectra can be attributed to the partial and random interactions among multiple small pulses within the PCDSB in Fig. 4(b). Furthermore, Fig. 4(c) summarizes and displays the intensity evolution over 1500 round trips from Fig. 4(a). The results indicate the presence of spiky features in the PCDSB, which are also reflected in the corresponding average spectrum. These features arise from the jitter of the PCDSB. Figure 4(c) shows a time interval of $\mathrm{\Delta }\tau =2.3\mathrm{ns}$ between the PCDSB and its left-adjacent wave, which appears only on the right side of the PCDSB in the OSA spectrum. Calculation shows that the wavelength separation between their peaks is $\mathrm{\Delta }\lambda =11.9\mathrm{nm}$, confirming that the adjacent wave is the single-peak structure near $\sim 1543.2\mathrm{nm}$ in the OSA spectrum (see Fig. 2).

Since the single-peak structure and the PCDSB are located on opposite sides of the zero-dispersion wavelength ($\sim 1533\mathrm{nm}$) of the HNLF in the OSA spectrum, we hypothesize that the long-wavelength portion of the composite structure consists of DWs generated by the PCDSB. To verify this hypothesis, the phase-matching condition for DWs in the HNLF must be satisfied [42]. The second- and third-order dispersion coefficients (${\beta }_{21}$ and ${\beta }_{31}$) for the PCDSB at $\sim 1531.3\mathrm{nm}$ are $0.044{\mathrm{ps}}^2/\mathrm{km}$ and $0.028{\mathrm{ps}}^3/\mathrm{km}$, respectively. We can roughly estimate the peak power of the PCDSB in the cavity to be $P_n=2.3\mathrm{W}$, and applying the phase-matching formula yields ${\mathrm{\Omega }}_n^2{\beta }_{21}/2+{\mathrm{\Omega }}_n^3{\beta }_{31}/6=\gamma P_n/2=6{\mathrm{km}}^{-1}$, where ${\mathrm{\Omega }}_n={\omega }_n-{\omega }_d$ is the frequency separation between the NLP at ${\omega }_n$ and the DW at ${\omega }_d$, and $\gamma$ is the nonlinear coefficient of the HNLF. Thus, we conclude that the single-peak structure indeed contains DWs generated by the PCDSB at approximately 1543.2 nm.

Simultaneously, it can be observed from Fig. 4(a) that the adjacent wave exhibits clear interference phenomena, which are caused by the interaction of DWs generated by multiple pulses within the PCDSB, resulting in a stable state. The parallel synchronized propagation of the PCDSB and DWs indicates that they achieve GV matching near the zero-dispersion point of the HNLF [42,43]. For both the PCDSB and DWs, SPM and XPM interactions play a critical role in balancing their attraction and repulsion, thereby maintaining the strong synchronized propagation of the FGC. Energy changes of the DWs and PCDSB within the FGC are shown in Figs. 4(d) and 4(e), over 1500 cycles prior to collision and 2000 cycles after collision. The difference in the degree of energy fluctuations in Figs. 4(d) and 4(e) also reflects that the stability of PCDSB and DWs is adjusted to some extent due to energy flow during the collision. The collisions reduce the intensities of both the DWs and PCDSB to varying extents, and the intensity ratio of PCDSB to DWs increases from 1.5 to 3 before and after the collision. However, both component energies remain relatively stable during the non-collision periods, with no significant energy exchange observed.

To confirm the components of the SGC, we first extracted the temporal evolution and the TS-DFT evolution diagram for this structure over 8000 cavity cycles (from the 1000th to the 9000th round trip) within a 4 ns range on the right side of the $x$-axis in Figs. 3(a) and 3(c), as shown in Figs. 5(a) and 5(b). This compound has a cavity period of 59.7821 ns (corresponding to a group velocity of 0.205747 m/ns). For clarity, we redrew the 2D evolution diagram, aligning the SGC trajectory parallel to the $y$-axis. The intensities for 2500 cycles (from the 1000th to the 3500th round trip), 1000 cycles (from the 4000th to the 5000th round trip), and 2500 cycles (from the 6500th to the 9000th round trip) were summed, producing the red, green, and blue curves shown in Fig. 5(c).

Figures 5(b) and 5(c) clearly reveal the presence of two synchronized wavelength components within the SGC. The broader, lower-intensity wavelength component on the left corresponds to the CS at $\sim 1555.2\mathrm{nm}$ in the OSA spectrum. The time interval of $\mathrm{\Delta }\tau =1.98\mathrm{ns}$ between the sharp peaks of the two wavelengths corresponds to a wavelength separation of $\mathrm{\Delta }\lambda =9.9\mathrm{nm}$, which matches the wavelength interval between the multi-peak structure and the CS in the OSA spectrum. This indicates that the high-intensity wavelength components on the right correspond to the multi-peak structure near $\sim 1545.3\mathrm{nm}$. In the inset of Fig. 2, the region to the right of the red line reveals a series of narrow spectral pulses. Moreover, these wavelength components do not exhibit significant spectral broadening after stretching through TS-DFT. Therefore, it can be inferred that the pulses of the high-intensity wavelength components are NSP. Although the attraction and repulsion between the CS and NSP, which have similar amplitudes, compete, the CS and NSP trap each other via the XPM-induced potential well, forming an SGC in a stable, non-collapsing balance. Notably, the formation of the SGC is also facilitated by the FWM effect provided by the HNLF. The inset in Fig. 5(c) shows that the energy of the CS within a single SGC remains stable during intracavity operation, while the energy of the NSP undergoes significant enhancement only during collisions, with no substantial energy exchange between the two components.

Figures 5(d) and 5(e) present the 2D temporal evolution and TS-DFT evolution spectra for the 2000 cycles following the collision. Due to the interactions between dispersion waves and the effects of gain depletion and recovery in the cavity, multiple SGCs interact with each other to form a joint state with an approximate temporal interval of several hundred picoseconds. This joint state can be considered as an SGC complex, as shown in Fig. 5(e). The key difference compared to previous works is the additional wavelength dimension [44,45]. In contrast to several isolated SGCs, the SGC complex exhibits interactions between the dichromatic soliton pulses and NSPs due to XPM and dispersion wave interactions, which result in visible interference within the SGC complex. Summing the intensity over 2000 cycles, as shown in Fig. 5(f), clearly reveals that the energy of the SGC complex is significantly higher than that of the isolated SGCs, and the presence of the CS component is more easily observed.

Moreover, in Fig. 5(c), the green curve indicates that the time interval between the two wavelength components of the single SGC decreases to $\mathrm{\Delta }\tau =1.81\mathrm{ns}$ during the intracavity collision process, while the blue curve shows that the time interval recovers to $\mathrm{\Delta }\tau =1.96\mathrm{ns}$ after the collision. In Fig. 5(f), the time interval differences between the CSs and NSPs in several isolated SGCs are also observed to vary. This variation in the time interval is attributed to the unstable oscillation of the central wavelength position of the NSPs caused by the collision, as observed in the multi-peak portion of the middle region of the average spectrum.

To clearly demonstrate the presence of synchronized dual-wavelength components in the SGC, we recorded several shot-to-shot spectra at different moments by reducing the attenuation level of the FOA before the DCF (see Fig. 1). Since the FOA is positioned outside the cavity and functions solely as a power attenuator, it does not affect the intracavity soliton dynamics or the spectral evolution observed in the DFT spectra. The corresponding spectral results are presented in Figs. 6(a)–6(d), each exhibiting distinct characteristics. The results from Figs. 6(a)–6(d) also highlight the robustness of the SGC during intracavity operation, where the energy saturation makes the measured CS component clearer than in Fig. 3(c). Furthermore, the joint state of multiple SGCs can appear either as a loosely bound complex within a certain number of round trips after regeneration, as shown in Figs. 3(c) and 6(b), or as a loosely bound state with smaller intervals, where distinguishing individual components—especially the distinct NSPs—from the TS-DFT spectra becomes challenging, as depicted in Figs. 6(a), 6(c), and 6(d).

As previously discussed, we performed a detailed analysis of the formation and components of two dichromatic compounds, the SGC complex and FGC, which have different GVs. During their intracavity propagation, collisions between them are inevitable. The region inside the red dashed lines in Fig. 1 shows where these two compounds collide within the cavity, leading to complicated dynamical processes. The primary events during the collision involve the annihilation of the SGC complex, the survival of the SGC, and the regeneration of the SGC complex. Figure 7(a) provides an enlarged view of the collision process occurring between the 1500th and 6500th round trips, as shown in Fig. 3(c). Figure 7(b) presents the corresponding 2D evolution diagram. It should be noted that in Figs. 7(a) and 7(b), the NSP on the far right and the CS on the far left of the SGC complex appear to exhibit a certain round-trip delay when colliding with the FGC. This is caused by the walk-off of different wavelength components during the TS-DFT measurement. In reality, however, during the actual collision and annihilation process between the SGC complex and FGC, the interactions among the composite components occur simultaneously.

Although XPM can locally minimize the binding energy of SGCs, enhancing the robustness of the compound against perturbations, SGCs can also break free from their potential well and undergo annihilation if they acquire sufficient energy during a collision [46]. In this scenario, SGCs survive by becoming trapped in the potential well if they do not gain enough energy during the collision. Notably, one SGC within the SGC complex in Fig. 7(a) cleverly survives the collision, avoiding annihilation, while the several isolated SGCs on the right side of the $x$-axis in Fig. 5(c) are survivors from previous periodic collisions. As mentioned earlier, although some isolated SGCs do not directly participate in the collision, during the process, the gain of the NSPs in these survivors significantly increases, triggering unstable oscillations in the central wavelength of the NSPs. This, in turn, enhances the XPM attraction between the NSPs and CSs. However, this gain process is relatively short-lived and does not disrupt the equilibrium of the non-colliding SGCs. Simultaneously, the SPM effect induced by the gain of the NSPs in each isolated SGC during the collision [47] results in the emergence of a metastable state, with the NSPs exhibiting pulsating behavior during the brief collision period [see Figs. 7(a) and 7(b)]. This behavior appears as an S-shaped trajectory in the shot-to-shot spectra, a phenomenon commonly observed in all isolated SGCs.

Interestingly, the collision spectra reveal that the SGC complex, after annihilation, re-emerges following multiple cavity cycles. The reborn CSs exhibit significant spectral broadening in their early stages [see Fig. 7(b), white dashed box], which is reconstructed through the occurrence of the FWM effect in HNLF by the unannihilated PCDSB and DWs. At the initial stage of NSP regeneration, also based on FWM, changes in intracavity gain enhance the intensity of the DWs, triggering the Kerr effect, which slows the DWs and causes them to drift toward longer wavelengths, while the intensity of the DWs along their original trajectory decreases [see Fig. 7(b), yellow dashed box]. During the deceleration of the DWs, a transient intermediate state forms [see Fig. 7(b), yellow dashed box]. Due to the XPM-induced mutual self-trapping effect, this intermediate state is captured by the CSs as multiple NSPs, thereby completing the frequency transition process for NSPs regeneration. Certainly, the intermediate state during the transition from DWs to NSPs is brief and does not significantly affect the stability of the FGC. Throughout the entire collision process, the interplay of gain modulation, dispersion, XPM, and FWM effects collectively leads to the disruption and reconstruction of intracavity balance, enabling the SGC complex to revive from the collision.

We have also recorded the annihilation and regeneration processes of SGC complexes at different moments, as illustrated in Figs. 6(a)–6(d). Although some of the regenerated states of SGC complexes are accompanied by more closely bound characteristics and their regeneration processes differ from each other, the FWM and XPM effects both play an integral role. Additionally, the DWs become weaker along their original trajectory while transitioning to the NSPs. The results show that such annihilation and regeneration processes are prevalent in the cavity. We also conduct simulations to verify the nonlinear propagation characteristics of multi-wavelength pulses in the HNLF. The results indicate that the regeneration of the 1555.2 nm CS within the cavity is likely associated with the multi-wavelength FWM process induced by the HNLF. Moreover, the energy flow among the four different wavelengths in the regeneration processes may also be related to the change of phase relationship between the two compounds. These relevant explanations can be found in the simulation (see Appendix A). These results indicate that the regeneration of SGC complexes after collisions in the resonant cavity is closely linked to the FWM process provided by the HNLF.

The nonlinear interactions within multi-dimensional soliton compounds can reveal many unresolved mysteries in nonlinear optics. The expansion of the wavelength dimension in mode-locked fiber lasers enables solitons to form synchronized dichromatic and multichromatic compounds, driven by various combination mechanisms that are closely associated with dispersion control and XPM interactions. This work has induced changes in the dissipative balance between multiple nonlinearities, dispersion, gain, and loss in the PMLFL, leading to the formation of two dichromatic synchronized soliton compounds and their subsequent collisions, which exhibit intriguing dynamic behaviors, such as the regeneration of dichromatic synchronized soliton compounds. The temporal characteristics of the SGC differ from those of previously reported synchronized dual-wavelength pulses [16]. Multiple SGCs form loosely bound complexes within the laser cavity through medium-range interactions, and collide with asynchronous FGCs. It is important to note that the FWM effect and XPM-induced mutual self-trapping of compounds play a crucial role in the regeneration of SGC complexes. Meanwhile, the annihilation and survival probabilities of SGCs within the loosely bound complexes depend on whether the energy gained during the collisions can overcome the constraints of their potential well. Additionally, SGCs that survived previous collisions exhibit short-lived metastable pulsations in the shot-to-shot spectral evolution during the collision process.

During the brief collision process, the equilibrium within the dissipative cavity is disrupted and then reconstructed. This unique balance transition highlights the complexity of multi-wavelength soliton dynamics and the rich variety of soliton behaviors. In summary, our study offers new insights into the phenomena of multichromatic soliton collisions and the underlying dynamical mechanisms in dissipative systems. We believe that a deeper understanding of these dynamics will pave the way for advancing applications in multi-dimensional optical information processing and high-throughput optical communication.