# Non-Newtonian binary pulse interaction in a Mamyshev oscillator

Hervé Leblond, Florent Bessin, François Sanchez

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Hervé Leblond, Florent Bessin, François Sanchez. Non-Newtonian binary pulse interaction in a Mamyshev oscillator. Optics Communications, 2025, 574, pp.131164. 10.1016/j.optcom.2024.131164. hal-05529958

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# Non-Newtonian binary pulse interaction in a Mamyshev oscillator

Hervé Leblond *, Florent Bessin, François Sanchez

Laboratoire de Photonique d'Angers, EA 4464, Université d'Angers, 2 Boulevard Lavoisier, 49000 Angers, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Fiber laser

Mamyshev oscillator

Pulse interaction

## ABSTRACT

We consider the propagation and binary interactions of optical pulses in an anomalous dispersion Mamyshev oscillator modeled by means of nonlinear Schrödinger equations. Two eigenpulses with different profile and energy can propagate in the cavity. It is found that pulse interaction in the Mamyshev oscillator strongly differs from particle interaction through a potential: the interaction determines the relative pulse velocity, which remains unchanged when pulse separation varies.

## 1. Introduction

Since the pioneer theoretical study by Malomed [1], pulse interactions in fiber lasers have been the matter of extensive research, both experimentally and theoretically. Binary interactions were described in the frame of the complex Ginzburg-Landau (CGL) equation [2-4], and observed in fiber lasers mode-locked through the nonlinear polarization rotation technique [5-8] but also in figure-eight fiber lasers [9].

Bunches containing more than two pulses were early observed [10-12] and theoretically treated using binary interactions [3,13], but also ternary [14,15].

Later developments involved, from the experimental viewpoint, the interaction of large assemblies of pulses, which were compared to the states of matter [16], forming large structures as soliton crystals [17], or exotic behaviors such as soliton rain [18]. High-order harmonic mode-locking was observed [19], and explained both from repulsive binary interactions and full numerical simulation of the laser cavity [20].

Theory was interested in interactions of more complex structures, such as vortices [21], pulse interactions in presence of noise [22]. More recently experiment focuses on oscillating structures involving more than a bound pulse pair, called breather molecular complexes [23-25]. The detailed evolution of the interaction has become experimentally available, e.g. in the case of the collision between a pulse and a bound pair [26,27].

As early as 1994, Piché [28] proposed and numerically demonstrated the use of self-phase modulation (SPM) in a Kerr medium placed between two spectrally shifted filters in a laser cavity to achieve mode-locking. The idea, however, was not developed further before it reappeared in a rather different context, that of pulse regeneration. Mamyshev [29] showed that using a nonlinear medium inducing

SPM and a spectral filter can restore the quality of a signal. Systematic studies of the process followed this work [30]. The behavior of a pulse launched in a chain of regenerators was considered in 2008 [31], opening the way to use the device as a source [32,33]. Ultrashort pulses with a pulse duration of about 40 fs after compression and a record pulse peak power of about 1 MW were produced in 2017 [34], and subsequent papers showed increasing power and decreasing duration [35,36]. Multiple pulse dynamics in Mamyshev oscillator were soon observed both theoretically and experimentally. In [37] was shown that two different eigenpulses can coexist in the cavity. The first systematic experimental studied of multiple pulse dynamics was performed in 2020 [38]; bound states of two pulses were then observed, later three and four pulse states as well as a theoretical analysis of bound states was proposed [39]. Harmonic mode-locking has been experimentally evidenced [40,41]. Recently was experimentally and theoretically demonstrated that the Mamyshev oscillator can generate pulsating dissipative soliton bound states [42].

In the present paper, we study theoretically a Mamyshev oscillator with anomalous fiber dispersion. We show that two different eigenmodes exist in this cavity. Then we study the binary pulse interactions that can arise in the considered setup. Although bound states were already considered theoretically in the normal dispersion regime [39,42], they never were in the anomalous one, and no systematic theoretical analysis of them has been published.

## 2. The model

The setup is shown on Fig. 1. It is composed of two arms, each of which is composed of an amplifying fiber (gain fiber, GF), a passive fiber (single mode fiber, SMF), and a band-pass filter.

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: herve.leblond@univ-angers.fr (H. Leblond).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 1. The analyzed setup. GF: gain fiber. SMF: single mode fiber.</figcaption></figure>

The propagation in passive fibers is modeled through the nonlinear Schrödinger (NLS) equation, as

$$i \frac{\partial A}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + \gamma A |A|^2 = 0, \quad (1)$$

where A is the electric field amplitude, z the propagation distance, t the time in a frame of reference moving at the group velocity, $\beta_2 = -0.02 \text{ ps}^2/\text{m}$ the group velocity dispersion, and $\gamma = 1.2 \text{ W}^{-1} \text{ km}^{-1}$ the nonlinear coefficient. We used the parameters of an experiment performed in our lab. First trials tend to indicate that dispersion is essential and that a completely different behavior can be expected under normal dispersion. Propagation in the gain fibers is described by an analogous equation with additional gain term, namely

$$i \frac{\partial A}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 A}{\partial t^2} + \gamma A |A|^2 - iG * A = 0, \quad (2)$$

where the gain coefficient G is defined in the Fourier domain as

$$\hat{G} = \frac{G_0 e^{-(\omega/\omega_g)^2}}{1 + E/E_s}. \quad (3)$$

There $\omega$ is the Fourier variable of the NLS equation, i.e., the angular frequency shift with respect to the angular frequency of the carrier wave. The parameters $\omega_g = 10 \text{ rad ps}^{-1}$, $E = \int |A|^2 dt$, and $E_s = 0.3 \text{ nJ}$ are the spectral width of the amplifier, the pulse energy, and the saturation energy, respectively. In our approach, $E_s$ is assumed to be constant and chosen in a ad hoc way.

The unsaturated gain coefficient $G_0$ is adjusted in such a way that a stable one pulse steady state is formed. The range of values of $G_0$ that ensures this is rather small, and the values themselves decrease as the filter bandwidth $\Delta\lambda_{filter}$ increases (we use $G_0 = 0.75$, 0.6 and $0.5 \text{ m}^{-1}$ for $\Delta\lambda_{filter} = 2, 2.5$ and $3 \text{ nm}$ respectively). The length of each of the four fibers is fixed at $L = 5 \text{ m}$. The effect of the filters is accounted for by multiplying in Fourier space the field amplitude by the super-Gaussian filter transmission

$$F_{\pm} = F_0 \exp \left( - \left[ \frac{v \mp \delta v_{\text{off set}}}{\delta v_{\text{filter}}} \right]^m \right) \quad (4)$$

where $\Delta v_{filter}$ is the half spectral width of the filter, and $\delta v_{off set}$ its central frequency, relative to the carrier frequency. In the following, we use $m = 10$. The differential Eq. (1)-(2) are solved in the Fourier domain using a standard fourth-order Runge-Kutta scheme, with fast Fourier transforms at each step to compute the nonlinear term.

### 3. Eigenpulses

We fix the spectral width of the filter, as $\Delta\lambda_{filter} = 2, 2.5$ or $3 \text{ nm}$ (with $\Delta v_{filter} = c\Delta\lambda_{filter}/(2\lambda^2)$, where c is the speed of light in vacuum and $\lambda = 1.55 \text{ µm}$), and vary the distance $\delta\lambda_{off set}$ between the center frequency of the two filters (with the same relation as above between $\delta\lambda_{off set}$ and $\delta v_{off set}$). The temporal profile of a sample pulse, for $\Delta\lambda_{filter} = 3 \text{ nm}$ and $\delta\lambda_{off set} = 3.6 \text{ nm}$ recorded right after the gain fiber 1, after the passive fiber 1, and after upper band filter is shown in Fig. 2(a) with blue dashed line, red solid line, and black dotted line, respectively. The corresponding pulse spectrum is depicted in Fig. 2(b), using the same color code. Fig. 2(c) and (d) displays the same sampling

as for (a) and (b) but for second arm of Mamyshev oscillator. We get the narrowest pulse profile, for higher peak amplitude, and larger spectral width, just after the passive fibers (red curves). Right after filters we get a narrow spectrum with low peak power amplitude. Note that the spectral profile presents small sidebands, one of which is located at zero relative frequency, i.e., at the frequency of the carrier wave, when sampled just after the amplifier (blue curves).

We then studied the evolution of the pulse characteristics as the filter separation $\delta\lambda_{off set}$ varies. These evolutions are depicted in Fig. 3, right after the first amplifier for three values of the filter bandwidth, namely $\Delta\lambda_{filter} = 2, 2.5$ and $3 \text{ nm}$, with black dotted lines, blue dashed lines and red solid lines, respectively. When the bandpass of both filters overlaps, e.g., for $\Delta\lambda_{filter} = 3 \text{ nm}$, with $\delta\lambda_{off set} \le 2.86 \text{ nm}$, the continuous wave component grows considerably and largely dominates the pulse spectral amplitude. Such situation is excluded from the study.

Fig. 3a shows the peak power of the pulse versus $\delta\lambda_{off set}$. For each value of $\Delta\lambda_{filter}$, the curve consists in two branches, the upper branch being shorter than the lower one. The eigenstates are obtained from an arbitrary Gaussian seed and randomly belong to one branch or the other. To complete a branch, we use an eigenstate as initial data and slightly change the filter offset. It is clear that only stable eigenstates are found this way, unstable ones may also exist. The largest values of the peak power are obtained for the largest filter bandwidth ($\Delta\lambda_{filter} = 3 \text{ nm}$, red curve), on the upper branch, close to its right end, for $\delta\lambda_{off set}$ roughly equal to $1.1\Delta\lambda_{filter}$ (the exact values are 2.20, 2.82 and 3.42 nm for $\Delta\lambda_{filter} = 2, 2.5$ and $3 \text{ nm}$ respectively). The lower branch does not grow monotonically with the offset $\delta\lambda_{off set}$, although more peak power and energy are expected to be required to induce more self-phase modulation to overpass the filters when the offset is larger. The possibility to generate different eigenpulses in the cavity is consistent with the theoretical prediction of [31].

The corresponding pulse energy versus $\delta\lambda_{off set}$ is shown in Fig. 3b, using same color code as Fig. 3a. The range of variation of the energy with respect to the pulse separation $\delta\lambda_{off set}$ is rather small, while its variations against the bandwidth $\Delta\lambda_{filter}$ are much wider. Besides, the maxima are located roughly at the same places as for the amplitudes ($\delta\lambda_{off set} = 2.20, 2.82$ and $3.42 \text{ nm}$ for $\Delta\lambda_{filter} = 2, 2.5$ and $3 \text{ nm}$ respectively).

Finally, in Fig. 3c we show the pulse duration after the first passive fiber. It is computed as

$$\Delta t = \left\langle (t - \langle t \rangle)^2 \right\rangle^{1/2}, \quad (5)$$

where $\langle \cdot \rangle$ designates the mean value with weight I such as

$$\langle f \rangle = \frac{\int I f dt}{\int I dt}. \quad (6)$$

We note that $\Delta t$ strongly decreases as the filter bandwidth $\Delta\lambda_{filter}$ increases. For each value of $\Delta\lambda_{filter}$, the minimum of the duration $\Delta t$ is obtain at the end of the shortest branch, close to the point of maximal amplitude.

After the passive fiber, the pulse duration $\Delta t$ is only slightly shorter than before it.

The spectral width after the first passive fiber, is shown in Fig. 3d. The spectral width in terms of frequency $\Delta v_p$ is first computed in a way analogous to Eq. (5), and then converted in terms of wavelength as $\Delta\lambda_p$. We can note that, for example, when $\Delta\lambda_{filter} = 3 \text{ nm}$, the shortest pulse duration is about 1.2 ps while the spectral bandwidth is about 4.5 nm revealing that it is not transform limited. Assuming sech-pulse, the limit is about 560 fs and the peak power 8 kW. We however tried to compress numerically the pulses and found it impossible due to their phase profile.

Consistently with the variations of the duration depicted in Fig. 3c, the widest spectra are obtained for the largest filter bandwidth $\Delta\lambda$. For $\Delta\lambda = 2$ and $2.5 \text{ nm}$, the maxima are no longer located at the end of the short upper branch, but further on the long lower branch. After the first passive fiber (Fig. 3d), the spectral width has been multiplied by
<figure><img src="image_5.png" /><figcaption>Fig. 2. (a) The field amplitude for the first half of the mamyshev oscillator. Blue dashed line: after the amplifier; red solid line: after the passive fiber; black dotted line: after the filter. (b) The corresponding spectrum, with the same color code. (c) Same as (a), but for the second half of the mamyshev oscillator. (d) The corresponding spectrum. Obtained for filter bandwidth $\Delta\lambda_{\text{filter}} = 3$ nm and offset $\delta\lambda_{\text{offset}} = 3.6$ nm. The thin solid black lines on (b) and (d) show the filter transfer functions $F_{\pm}$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Just after the first passive fiber: (a) The pulse peak power, (b) the pulse energy, (c) the pulse duration, and (d) the pulse spectral width $\Delta\lambda_p$. Red solid lines: $\Delta\lambda_{\text{filter}} = 3$ nm; blue dashed lines: $\Delta\lambda_{\text{filter}} = 2.5$ nm, black dotted lines: $\Delta\lambda_{\text{filter}} = 2$ nm.</figcaption></figure>

at least 20. The factor increases with the filter bandwidth, becoming close to 30 for the upper branch with $\Delta\lambda = 3$ nm. This upper branch is markedly farther from the lower branch than it was after the amplifier.

## 4. Binary pulse interactions

We build an initial state by taking two numerically obtained steady-state pulses, either identical or both belonging to different branches, and set them at a fixed delay time from each other. The evolution of this input is then computed and recorded in the case where $\Delta\lambda_{\text{filter}} = 2.5$ nm. We first fix the initial time delay to be $\Delta t = 32$ ps, and vary the filter separation $\delta\lambda_{\text{offset}}$.

If the two filters overlap, i.e., if $\delta\lambda_{\text{offset}} < \Delta\lambda_{\text{filter}}$, more precisely if $\delta\lambda_{\text{offset}} \le 2.48$ nm, the stability of the input pulses is lost, and they turn

to a continuous wave. For $\delta\lambda_{\text{offset}} \ge 2.49$ nm this scenario is impossible, since the continuous wave cannot propagate.

Destabilization however still occurs, and a single excited pulse can be obtained. This behavior is depicted in a pseudo-color plot in Fig. 4 where we show the temporal profile versus the roundtrip number obtained from numerical simulations. We note that this pulse suffers rather large amplitude and position random variations. In addition, it varies randomly between one and two-humped profiles, recalling somehow a second-order soliton, but with no regularity. This outcome is observed, if the two input pulses belong to the lower branch, up to $\delta\lambda_{\text{offset}} = 2.74$ nm. Before this value, the instability strongly slows down: it occurs at 52 round-trips for $\delta\lambda_{\text{offset}} = 2.7$ nm, 215 round-trips for 2.72 nm, 454 round-trips for 2.73 nm. For two pulses belonging to the upper branch, the upper boundary is $\delta\lambda_{\text{offset}} = 2.77$ nm, while if one pulse belongs to each branch, it is $\delta\lambda_{\text{offset}} = 2.74$ nm.
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Pulse destabilization leading to a single noisy pulse with a lot of jitter. For one pulse on each branch, filter separation $\delta\lambda_{offset} = 2.7$ nm and initial time delay $\Delta t = 32$ ps. (a) Evolution, (b) temporal profile, (c) spectrum. Blue dashed line: roundtrip number $n = 10$; red solid line: $n = 29$; black dotted line: $n = 36$.</figcaption></figure>

For higher values of $\delta\lambda_{offset}$, a regime with random amplitudes and separation fluctuations, but conserving the two pulses, is seen. An example is depicted in Fig. 5 for both pulses belonging to the lower branch for $\delta\lambda_{offset} = 2.76$ nm, but can be observed for $\delta\lambda_{offset}$ in the range from 2.75 nm to 2.76 nm. This behavior can also be observed when pulses belong to different branches in the same range of filter offset $\delta\lambda_{offset}$, but was not observed for both pulses on the upper branch.

By further increasing the filter separation $\delta\lambda_{offset}$, another regime is reached, in which both pulses move with the same constant velocity, which differs from their own speed. We illustrate this case in Fig. 6 for $\delta\lambda_{offset} = 2.9$ nm and both pulses on the upper branch. In this example the pulse speed was, when it was alone, $v_0 = 0.0284$ ps turn$^{-1}$, while it becomes $v = 0.0454$ ps turn$^{-1}$ for the pulse pair. It can be seen from Fig. 6c that the oscillations of the spectrum do not move as the pulse pair evolves, which proves that the pulses are phase-locked, and hence form a bound state. This behavior extends from $\delta\lambda_{offset} = 2.77$ nm to 2.89 nm.

This is also observed for two pulses on the upper branch, and happens for $2.77$ nm $\le \delta\lambda_{offset} \le 2.86$ nm, which is the end of the upper branch. When pulses belong to different branches this occurs for $2.77$ nm $\le \delta\lambda_{offset} \le 2.79$ nm. For the lowest values of $\delta\lambda_{offset}$ in this regime, small regular amplitude oscillations are observed.

<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Random time delay fluctuations of the interacting pulses. For both pulses on the lower branch, filter separation $\delta\lambda_{offset} = 2.76$ nm and initial time delay $\Delta t = 32$ ps. (a) Evolution, (b) temporal profile, (c) spectrum. Blue dashed line: roundtrip number $n = 100$; red solid line: $n = 500$; black dotted line: $n = 900$.</figcaption></figure>

For values of $\delta\lambda_{offset}$ between 2.9 nm and 3.02 nm, when both pulses belong to the lower branch, the pulses repel each other, taking constant opposite velocities (see Fig. 7). The speed at which the pulses move away from each other decreases when $\delta\lambda_{offset}$ increases. If pulses belong to different branches, repulsion occurs from $\delta\lambda_{offset} = 2.8$ nm to 2.86 nm, but if both pulses belong to the upper branch, repulsion is not observed.

For values of $\delta\lambda_{offset}$ of 3.03 nm and higher, which requires that both pulses belong to the lower branch, the initial slow repulsion soon relaxes to a zero velocity, as shown in Fig. 8. The direction of the transient is changed to an attraction between 3.1 nm and 3.38 nm, and above 3.52 nm, but the outcome is the same: immobility, with a separation distance essentially depending on the initial one. In contrast with the interaction observed in fiber lasers mode-locked by means of the nonlinear polarization rotation technique, or in the frame of the CGL equation, there is no equilibrium distance, although a steady state is reached. As a result, and still in contrast with what has been observed in other setups, there is no quantization of the distance between pulses in a bound state.

For values of $\delta\lambda_{offset}$ of 3.61 nm and larger, the destabilization towards a single noisy pulse with considerable jitter occurs as it did below 2.73 nm, but it leads more and more quickly to the extinction of the laser. It is seen that the outcome only marginally depends on
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 6.** The two interacting pulses move with identical velocity. For both pulses on the upper branch, filter separation $\delta\lambda_{offset} = 2.82$ nm and initial time delay $\Delta t = 32$ ps. (a) Evolution, (b) temporal profile, (c) spectrum. Blue dashed line: roundtrip number $n = 1$; red solid line: $n = 250$; black dotted line: $n = 450$.</figcaption></figure>

the branch to which the pulses belong. Repulsion, however, was not observed on the upper branch.

A more systematic investigation is performed, with varying not only the filter separation $\delta\lambda_{offset}$, but also the initial pulse time delay $\Delta t$. On the other hand, the study is restricted to the case where both pulses belong to the lower branch, the longer one. The result is shown in Fig. 9 where two important features are to be noticed. The first one is the arising of a further type of interaction, where pulses attract each other with a constant velocity. When the pulses become too close together, the velocities invert and interaction turns to repulsion (see Fig. 10). The minimal distance is about $\Delta t = 25$ ps in every computed case. It can be seen that each pulse posses a pedestal, and that repulsion starts as the pedestals come in contact one to each other: both pulses are merely colliding (see Fig. 10b). The second one, except the special situation above, is that the interaction type and outcome appreciably depend on the initial delay time, while, as soon as it is started, the velocity does not depend on the delay time any more, but remains constant all along the propagation. For the lower values of $\delta\lambda_{offset}$, the two-pulse input decays to a continuous wave (domain 1), while laser extinction is observed for the larger ones (domain 8). Pulse pairs are formed in two ways, either relaxation to a vanishing velocity in the wide domain 7, or two pulses moving with identical velocity in domain 4. In the latter case, there is no relaxation and the velocities are identical from

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 7.** The two interacting pulses repel each other with constant velocity. For both pulses on the lower branch, filter separation $\delta\lambda_{offset} = 2.9$ nm and initial time delay $\Delta t = 32$ ps. (a) Evolution, (b) temporal profile, (c) spectrum. Blue dashed line: roundtrip number $n = 1$; red solid line: $n = 500$; black dotted line: $n = 999$.</figcaption></figure>

the very beginning. In between attraction (domain 6) and repulsion (domain 5) occur, depending largely on the initial pulse separation. As said above, this attraction does not lead to stabilization into a pulse pair but eventually turns to repulsion. We cannot affirm that long-range interaction exists since the pulse acquire a group velocity from the very beginning, and merely conserve it afterwards. For low values of $\delta\lambda_{offset}$, the pair decays to a single noisy pulse (domain 2). It must be noticed that the extension of this domain is wider for the shortest initial pulse separations: this issue replaces pulse pair formation if the initial distance is too short. In the same way, laser extinction also occurs for short input pulse separations. In the narrow transition domain 3 the distance between the pulses randomly fluctuates.

## 5. Conclusions

We considered pulse propagation in a Mamyshev oscillator modeled by means of NLS equations. Two eigenpulses with different profile and energy can propagate in the cavity. The pulse interaction in the Mamyshev oscillator is not Newtonian, in the sense that it does not resemble particle interaction through a potential at all. When attraction or repulsion occurs, the velocity does not change as the pulses become closer or more far away. When one pulse disappears, there is no collision, it vanishes at its initial location. Coexistence of two pulses
<figure><img src="image_15.png" /><figcaption>Fig. 8. The two interacting pulses relax to zero velocity. For both pulses on the lower branch, filter separation $\delta\lambda_{offset} = 3.34$ nm and initial time delay $\Delta t = 32$ ps. (a) Evolution, (b) temporal profile, (c) spectrum. Blue dashed line: roundtrip number $n = 1$; red solid line: $n = 250$; black dotted line: $n = 499$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 9. The various outcomes of the binary pulses interaction versus the filter separation $\delta\lambda_{offset}$ and the initial time delay $\Delta t$. Red crosses (1): decay to continuous wave; blue circles (2): decay to a single noisy pulse; cyan stars (3): time delay fluctuations; magenta diagonal crosses(4): both pulses have identical constant velocities; red squares(5): the pulses repel each other with constant velocity; blue down-pointing triangles (6): the pulses attract each other with constant velocity; green up-pointing triangles (7): relaxation to a state with zero velocity; black right-pointing triangles (8): laser extinction. Black lines are a mere guide for the eye.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 10.** The two interacting pulses attract each other with constant velocity, until a minimal time delay of about 25 ps is reached, then repel each other with constant velocity. For both pulses on the lower branch, filter separation $\delta\lambda = 2.98$ nm and initial time delay $\Delta t = 53$ ps. (a) Evolution, (b) temporal profile at minimal separation distance, $n = 634$.</figcaption></figure>

with fixed time delay, either moving together or fixed, is quite frequent, but never implies an equilibrium distance: in many cases it remains unchanged, and always depends essentially of its initial value. As a result, pulse separation in pulse pairs is not quantified. This situation would allow stable multiple pulse operation of the laser, in which the distance between pulses changes random from one pulse pair to another, but does not vary with time. When attraction or repulsion occur, each pulse will move permanently form one neighbor to the other with constant speed, changing direction at the collisions. It must also be paid attention to the fact that for some values of the filter separation multiple pulsing merely renders the laser unstable.

### CRediT authorship contribution statement

**Hervé Leblond:** Writing – original draft, Investigation. Florent Bessin: Writing – original draft, Investigation. **François Sanchez:** Writing – original draft, Investigation.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### References

[1] B.A. Malomed, Bound solitons in the nonlinear Schrödinger–Ginzburg–Landau equation, Phys. Rev. A 44 (1991) 6954.

[2] V.V. Afanasjev, B.A. Malomed, P.L. Chu, Stability of bound states of pulses in the Ginzburg–Landau equations, Phys. Rev. E 56 (1997) 6020.

[3] N.N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Multisoliton solutions of the complex Ginzburg–Landau equation, Phys. Rev. Lett. 79 (1997) 4047.

[4] N.N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Stable soliton pairs in optical transmission lines and fiber lasers, J. Opt. Soc. Am. B 15 (1998) 515.

[5] D.Y. Tang, W.S. Man, H.Y. Tam, P.D. Drummond, Observation of bound states of solitons in a passively mode-locked fibre laser, Phys. Rev. A 64 (2001) 033814.

[6] D.Y. Tang, B. Zhao, D.Y. Shen, C. Lu, W.S. Man, H.Y. Tam, Bound-soliton fiber laser, Phys. Rev. A 66 (2002) 033806.

[7] Ph. Grelu, F. Belhache, F. Gutty, J.M. Soto-Crespo, Phase-locked soliton pairs in a stretched-pulse fiber laser, Opt. Lett. 27 (2002) 966.

[8] A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, F. Sanchez, Ultra-short bound states generation with a passively mode-locked high-power Yb-doped double-clad fiber laser, Opt. Comm. 225 (2003) 71–78.

[9] N.H. Seong, D.Y. Kim, Experimental observation of stable bound solitons in a figure-eight fibre laser, Opt. Lett. 27 (2001) 1321.

[10] Y. Gong, P. Shum, T. Hiang, Cheng, Q. Wen, D.Y. Tang, Bound solitons pulses in passively mode-locked fibre laser, Opt. Commun. 200 (2001) 389.

[11] F. Gutty, Ph. Grelu, N. Huot, G. Vieene, G. Millot, Stabilisation of modelocking in fibre ring laser through pulse bunching, Electron. Lett. 37 (2001) 745.

[12] B. Ortaç, A. Hideur, T. Chartier, M. Brunel, Ph. Grelu, H. Leblond, F. Sanchez, Generation of bound states of three ultrashort pulses with a passively mode-locked high-power Yb-doped double-clad fiber laser, IEEE Photon. Technol. Lett. 16 (5) (2004) 1274–1276.

[13] X. Liu, Interaction and motion of solitons in passively-mode-locked fiber lasers, Phys. Rev. A 84 (2011) 053828.

[14] N. Akhmediev, J.M. Soto-Crespo, M. Grapinet, Ph. Grelu, Dissipative soliton interactions inside a fiber laser cavity, Opt. Fiber Technol. 11 (2005) 209–228.

[15] H. Leblond, A. Komarov, M. Salhi, A. Haboucha, F. Sanchez, “CIS” bound states of three localized pulses of the cubic-quintic CGL equation, J. Opt. A: Pure Appl. Opt. 8 (2006) 319–326.

[16] F. Amrani, A. Haboucha, M. Salhi, H. Leblond, A. Komarov, F. Sanchez, Dissipative solitons compounds in a fiber laser. Analogy with the states of the matter, Appl. Phys. B. 99 (1–2) (2010) 107–114.

[17] A. Haboucha, H. Leblond, M. Salhi, A. Komarov, F. Sanchez, Analysis of soliton pattern formation in passively mode-locked fiber lasers, Phys. Rev. A 78 (2008) 043806.

[18] S. Chouli, Ph. Grelu, Soliton rains in a fiber laser: An experimental study, Phys. Rev. A 81 (2010) 063829.

[19] F. Amrani, A. Haboucha, M. Salhi, H. Leblond, A. Komarov, P. Grelu, F. Sanchez, Passively mode-locked erbium-doped double-clad fiber laser operating at the 322nd harmonic, Opt. Lett. 34 (14) (2009) 2120–2122.

[20] A. Komarov, H. Leblond, F. Sanchez, Passive harmonic mode-locking in a fiber laser with nonlinear polarization rotation, Opt. Comm. 267 (2006) 162–169.

[21] D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, B.A. Malomed, Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg–Landau equation, Phys. Rev. A 77 (2008) 033817.

[22] O. Descalzi, J. Cisternas, D. Escaff, H.R. Brand, Noise induces partial annihilation of colliding dissipative solitons, Phys. Rev. Lett. 102 (2009) 188302.

[23] J. Peng, Z. Zhao, S. Boscolo, Ch. Finot, S. Sugavanam, D.V. Churkin, H. Zeng, Breather molecular complexes in a passively mode-locked fiber laser, Laser Photon. Rev. 15 (2021) 2000132.

[24] Y. Zhou, J. Shi, Y.-X. Ren, K.K.Y. Wong, Reconfigurable dynamics of optical soliton molecular complexes in an ultrafast thulium fiber laser, Commun. Phys. 5 (2022) 302.

[25] Dichromatic breather molecules in a mode-locked fiber laser, Y. Cui, Y. Zhang, L. Huang, A. Zhang, Z. Liu, C. Kuang, C. Tao, D. Chen, X. Liu, B.A. Malomed, Phys. Rev. Lett. 130 (2023) 153801.

[26] J. He, P. Wang, R. He, C. Liu, M. Zhou, Y. Liu, Y. Yue, B. Liu, D. Xing, K. Zhu, K. Chang, Z. Wang, Elastic and inelastic collision dynamics between soliton molecules and a single soliton, Optics Express 30 (9) (2022) 14218.

[27] J. Li, H. Li, Z. Wang, Z. Zhang, S. Zhang, Y. Liu, Real-time access to collisions between a two-soliton molecule and a soliton singlet in an ultrafast fiber laser, Photonics 9 (2022) 489.

[28] Michel Piché, Mode-locking through nonlinear frequency broadening and spectral filtering, Proc. SPIE 2041 (1994) 358–365.

[29] P.A. Mamyshev, All-optical data regeneration based on self-phase modulation effect, in: ECOC'98, IEEE, 1998, pp. 475–476.

[30] L. Provost, C. Finot, P. Petropoulos, K. Mukaza, D.J. Richardson, Design scaling rules for 2R-optical self-phase modulation-based regenerators, Opt. Exp. 15 (2007) 5100.

[31] S. Pitois, C. Finot, L. Provost, Asymptotic properties of incoherent waves propagating in an all-optical regenerators line, Opt. Lett. 15 (2007) 3263–3265.

[32] M. Rochette, L.R. Chen, K. Sun, J. Hernandez-Cordero, Multiwavelength and tunable self-pulsating fiber cavity based on regenerative SPM spectral broadening and filtering, IEEE Phot. Tech. Lett. 20 (2008) 1497–1499.
[33] K. Sun, M. Rochette, L.R. Chen, Output characterization of a self-pulsating and aperiodic optical fiber source based on cascaded regeneration, Opt. Express 17 (2009) 10419–10432.

[34] Z. Liu, Z.M. Ziegler, L.G. Wright, F. Wise, Megawatt peak power from Mamyshev oscillator, Optica 4 (2017) 649–654.

[35] W. Liu, R. Liao, J. Zhao, J. Cui, Y. Song, C. Wang, M. Hu, Femtosecond Mamyshev oscillator with 10-MW-level peak power, Optica 6 (2019) 194–197.

[36] C. Ma, A. Khanolkar, Y. Zang, A. Chong, Ultrabroadband, few-cycle pulses directly from a Mamyshev fiber oscillator, Photon. Res. 8 (2020) 65–69.

[37] S. Pitois, C. Finot, L. Provost, D.J. Richardson, Generation of localized pulses from incoherent wave in optical fiber lines made of concatenated Mamyshev regenerators, J. Opt. Soc. Am. B 25 (2008) 1537–1547.

[38] S.S. Xu, M. Liu, Z.W. Wei, A.P. Luo, W.C. Xu, Z.C. Luo, Multipulse dynamics in a Mamyshev oscillator, Opt. Lett. 45 (2020) 2620–2623.

[39] D. Yan, X. Li, S. Zhang, J. Liu, Pulse dynamic patterns in a self-starting Mamyshev oscillator, Opt. Exp. 29 (2021) 9805–9815.

[40] E. Poeydebat, F. Scol, O. Vanvincq, G. Bouwmans, E. Hugonnot, All-fiber Mamyshev oscillator with high average power and harmonic mode-locking, Opt. Lett. 45 (2020) 1395–1398.

[41] Y. Sui, L. Jin, Z. Liu, L. Tao, H. Zhang, M. Bi, X. Zhou, High signal-to-noise ratio harmonic mode-locking Mamyshev oscillator at 1550 nm, Opt. Commun. 569 (2024) 130787.

[42] B. Cao, K. Zhao, C. Gao, X. Xiao, C. Bao, C. Yang, Observation of pulsating dissipative solitons in a Mamyshev oscillator, Phys. Rev. A 106 (2022) 023519.