Interactions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order are numerically investigated in optical fibers. When the soliton order is 1, depending on different initial pulse intervals, the initial in-phase Airy pulses may evolve to single breathing solitons, bound breathing solitons, and single parallel breathing solitons. While the out-of-phase Airy pulses may evolve to parallel or repulsive soliton pairs with breathing or weak breathing, after radiating away some dispersive waves. When the initial relative phases take arbitrary values except 0 and π, moving single breathing solitons and repulsive or parallel soliton pairs will form. Moreover, the whole temporal profiles may become asymmetric. The repulsive soliton pairs consist of two moving breathing solitons with different intensities, moving velocities, and breathing periods. The most interestingly is that, when the soliton order is larger than one, we observe double bound breathing solitons, double parallel breathing soliton pairs, and diverse composite breathing solitons which consist of two or more different breathing solitons. one can effectively manipulate and select the soliton expected and its evolution dynamics by adjusting the soliton order, initial pulse intervals, and initial relative phases.
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The optical soliton is all along a research hot spot for its significance both in theory and engineering applications. In particular, many special optical solitons with special profiles and unique features have been revealed recently. Typically, for instance, people have recently discovered optical rogue wave solitons , noise-like pulses , bound solitons , self-accelerating beams [4–10], and so on. Among them, self-accelerating beams, Airy beams, for instance, has intrigued extensive research interests since the first prediction of Airy wave packets in quantum mechanics in 1979  and then its theoretical introduction and experimental demonstration in optics in 2007 [5, 6]. Its tempting unique propagation features, such as self-healing, trajectory self-bending (or self-acceleration), and weak diffraction, allow for many potential applications typically including optical micro-manipulation [7–9], curved plasma channel generation , vacuum electron acceleration , supercontinuum generation , curved laser filamentation , optical routing , and so on. Therefore, investigation on Airy beams has been extended considerably in terms of the experimental realization , the propagation properties of single Airy beam or interactions among two or multiple Airy beams for different contexts [16–29], and various potential applications. Even only the propagation media involved have extended from initial linear media to current nonlinear ones, from initial local nonlinear media to current nonlocal ones [18–20], from ordinary media to chiral media , Bose-Einstein condenstates , the metal surface , photonic crystals , photorefractive crystal , and nematic liquid crystals , photonic lattices , atom vapors [28, 29], and turbulent media . Moreover, the optical beams involved are also modified from ordinary Airy beams to Airy-Gaussian beams  and from complete coherent optical fields to partially coherent ones [30, 31]. Based on these works, many unique propagation properties and optical solitons are revealed. Typically, for example, it is discovered that, in the Kerr nonlinear media, the two Airy beams may evolve to single breathing solitons, or parallel, repulsive, or bound breathing soliton pairs, depending on different intervals and the relative phases between the two initial Airy beams . More interestingly, it is numerically demonstrated that, the nonlocal nonlinearity can support periodic intensity distribution of Airy beams with opposite bending directions and stable bound states of both in-phase and out-of-phase breathing Airy solitons which always repel in local nonlinear media . Furthermore, researchers have also revealed the nonparaxial self-accelerating beams such as Mathieu beams and Weber beams which can move along circular, elliptical, and parabolic trajectories with large angle bending but still retain nondiffracting and self-healing properties .
Correspondingly, the temporal domain counterparts of the self-accelerating beams–Airy pulses have also intrigued special research interests due to their bending temporal trajectories, high dispersion resistibilities, and self-healing properties in the linear dispersion media [33, 34]. Meanwhile, researchers pay much attention to their various nonlinear propagation phenomena such as the time-domain inversion and tight focusing , supercontinuum generation , soliton shedding , nonlinear spectral shaping , and so on. Moreover, influence of a variety of higher-order effects such as quintic nonlinearity [38, 39], third-order dispersion , self-steepening , and Raman scattering , on the propagation dynamics of Airy pulses are also studied. In comparison with the spatial case, however, interactions of Airy pulses and their corresponding new types of solitons generation have rarely been reported . In , the authors have explored the dynamical propagation of in-phase Airy pulses with and without initial chirps for short propagation distance in case of small number of pulse intervals. Naturally, our questions come and include the following two aspects. The first is that, how about the propagation dynamics of two Airy pulses for long distance. As for the second one, as demonstrated in  that, the interactions of the two Airy beams and the corresponding soliton characteristics are quite different for the initial relative phases and the intervals of two Airy beams. Thus, we may reasonably ask how the arbitrary initial relative phases and the intervals of the two Airy pulses influence their interactions and propagation dynamics. The third question is that, when the soliton order is large, whether the Airy pulses will collapse or evolve to new types of solitons. To make these things clear, we numerically investigate the interactions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order. The results reveal some interesting new types of solitons.
2. Propagation model
The standard nonlinear Schrödinger equation describing the optical propagating in the anomalous dispersion regime of an optical fiber is of the following form 
The incident pulse is the finite energy Airy pulse pairs, which is express as
3. Calculations and discussions
Based on Eqs. (1)-(4), we can numerically calculate the propagation dynamics of the Airy pulse pairs for different values of q0, N, and g by utilizing the split-step Fourier algorithm. We firstly discuss the case of N = 1. To observe more clearly the weak sub-pulses with lower intensity, we display the evolution of the normalized amplitude │u│ instead of the normalized intensity │u│2. In Figs. 1(a1)-1(h2), we show the contour maps of temporal evolutions of two in-phase and out-of-phase finite energy Airy pulses. Obviously, when the initial pulse intervals are large, the interactions of the two Airy pulses are weak, then we can observe parallel breathing solitons pairs generation whether the initial relative phase is 0 or π, as shown in Figs. 1(a1), 1(j1), 1(a2), and 1(h2). For the case of in-phase case g = 0, if q0 is negative, the two pulses overlap at certain weak side lobes. With decrease of the absolute value of minus q0, the pulse interval decreases and the overlapping energy increases. Correspondingly, single breathing solitons can gradually form after radiating some energy and the soliton intensities increase. However, the breathing periods shorten. The single breathing solitons can also be observed in cases of q0 = 0, 1, and 3. It is worth mentioning that although the biggest overlapping of two main lobs locate at about q0 = −1, the new formed initial pulse after superposing has relatively narrow and intense main lobe but weak side lobes compared to those of q0 = 0 and 1. Thus, the biggest interaction occurs at about q0 = 0 rather than q0 = −1 and the corresponding breathing period of the formed soliton is also the shortest. For the moderate values of positive q0, the bound breathing solitons will form, as shown in Figs. 1(g1)-1(i1). The bound breathing solitons consist of two breathing solitons which periodically attract and repel each other. Accordingly, the bound breathing solitons interestingly exhibit two forms of breathing. The whole breathing period increases with q0 whereas the sub-breathing one takes the opposite. While the out-of-phase Airy pulses can only evolve to parallel or repulsive soliton pairs after radiating away some dispersive waves. The parallel solitons are breathing whereas the repulsive soliton pairs are weak breathing or even hardly breathing. Besides the inner intense repulsive soliton pair, there are virtually more than one pair of repulsive dispersive waves outside. In comparison, these dispersive waves are weak but their repulsions are stronger. Moreover, they will disperse away gradually and disappear eventually. Besides, In Fig. 1(b2), the inner intense repulsive soliton pair comes from the fourth side lobes whereas the outer adjacent two dispersive waves respectively come from the third and second side lobes. While the outer last one pair of dispersive waves comes from the main lobes. For q0 = −5, we observe that the inner soliton pair comes from the third side lobes and the outer two dispersive waves respectively come from the main lobes and the second side lobes (not shown here). While in Figs. 1(c2)-1(g2), the inner intense soliton pairs come from the main lobes. However, it is worth mentioning that, in Figs. 1(c2)-1(e2), the initial main lobes actually generate from the destructive interference of the initial two Airy pulses. It can also be seen that, with decrease of the pulse interval, the interactions of the two pulse components become strong and the repulsions strengthen correspondingly. In addition, the formed soliton profiles in Figs. 1(a1)-1(h2) are all symmetrical with respect to . Obviously, the interactions of the in-phase and out-of-phase Airy pulses and the formed solitons characteristics here are quite similar to those of their spatial counterparts . It is not difficult to understand this similarity for the reason that the temporal dispersion and the spatial diffraction are in analogy with each other in terms of governing light propagation.
Besides the in-phase and out-of-phase cases, we also numerically calculate the evolutions of two Airy pulses with arbitrary relative phases. As typical examples, we display the evolution dynamics in cases of g = π/2 and π/3 in Figs. 1(a3)-1(i4). Clearly, the parallel breathing solitons can also form for the case of large pulse intervals, as shown in Figs. 1(a3), 1(i3), 1(a4), and 1(i4). However, in comparison with those in Figs. 1(a1)-1(h2), the parallel breathing solitons here consist of two different solitons in terms of their breathing periods and pulse intensities. For the cases of small and moderate pulse intervals, moving solitons can generate after radiating some dispersive waves. Their moving trajectories are straight lines. Depending on different relative phases and pulse intervals, the moving breathing solitons may consist of one or two moving breathing solitons and are also different in terms of their moving velocities, moving directions, breathing periods, pulse intensities, and pulse widths. It should be noted that the moving breathing soliton pairs here are quite similar to the repulsive soliton pairs in Figs. 1(a1)-1(h2). However, the latter obviously consists of two completely equivalent breathing solitons except for their different moving directions whereas the former consists of two different moving breathing solitons in terms of their moving velocities, moving directions, breathing periods, and pulse intensities. In addition, being different from those in Figs. 1(a1)-1(h2), all of the formed soliton profiles here are unsymmetrical with respect to . We have also tried calculating the evolution dynamics for the cases of other arbitrary relative phases and obtained similar results.
To observe more clearly and comprehensively the breathing characteristics of these solitons, we further calculate their maximal normalized amplitudes Um and the corresponding temporal positions , as displayed in Figs. 2 and 3 respectively. In the figures, L and R respectively stand for the soliton pulses which locate at the pulse leading and trailing edges. Obviously, all of Um curves oscillate damply surrounding certain average values except for that of the bound soliton whose parameters satisfy g = 0 and q0 = 6. The curves are different in terms or their oscillation periods, oscillation amplitudes, and average values, depending on different parameters g and q0. Except for the cases of g = 0 and π, the two pulses in the pulse pairs exhibit different values of Um. Specially, for the bound solitons, sub-oscillations can be observed superposing on the main oscillations [Figs. 2(b)]. And longer main oscillation periods correspond to larger initial pulse intervals. From Fig. 3, one can observe clearly three types of temporal trajectories. They are respectively damply oscillating curved trajectories for parallel soliton pairs [Fig. 3(a)], ordinary oscillating curved ones for bound breathing solitons [Fig. 3(b)], and straight ones for single moving breathing solitons [Fig. 3(c)] and unsymmetrical [Fig. 3(d)] or symmetrical [Fig. 3(e)] repulsive soliton pairs.
Secondly, we turn our attention to the case of N ≠ 1. In Fig. 4, we display the contour maps of interactions of two in-phase truncated Airy pulses for different pulse intervals and different soliton orders. When N is too small, no solitons form. In this case, after attracting and focusing at a certain distance, owing to the dominant dispersion effect, the central pulse broadens rapidly after radiating some energy in the form of repulsive dispersive waves, as shown in Figs. 4(a1) and 4(b1) where N = 0.3. For a little larger N but still smaller than 1, taking N = 0.8 for example, single breathing solitons with large breathing periods can generate. But the formed solitons are weak and wide, as shown in Figs. 4(c1) and 4(d1). When N is larger than 1 but still small, such as N = 1.5 and 2, single breathing solitons [Fig. 4(g1)], bound breathing solitons [Figs. 4(e1), 4(f1), 4(h1), and 4(k1)-4(m1)], and single parallel breathing soliton pairs [Figs. 4(i1), 4(j1), and 4(n1)], can still be observed, which is quite similar to the case of N = 1. Further increase of N may lead to generation of double parallel breathing soliton pairs or various composite breathing solitons besides the three kinds of breathing solitons mentioned above. The double parallel breathing soliton pairs consist of two pairs of parallel breathing solitons with different intensities, as shown in Figs. 4(u1), 4(z1), 4(f2), 4(g2), 4(m2), and 4(n2). In comparison, the outer breathing soliton pair is generally more intense than the inner one. However, one may notice that, for larger N, the single breathing solitons [Figs. 4(f2) and 4(g2)] or bound breathing solitons [Figs. 4(m2) and 4(n2)] may be sandwiched between the two breathing soliton pairs. But the sandwiched solitons are very weak in intensities and almost invisible. As for the composite breathing solitons, they consist of two or more than two kinds of breathing solitons. Typically, for example, the composite solitons in Figs. 4(p1), 4(x1), 4(d2), and 4(j2), consist of bound breathing solitons outside and single breathing soliton inside. Depending on different parameters N and q0, they exhibit different breathing periods and temporal widths. While in Fig. 4(y1), one can observe two bound breathing solitons with different breathing periods, intensities, and temporal widths. In Figs. 4(r1)-4(t1), weak bound breathing solitons are embed in two intense parallel breathing solitons with their temporal trajectories slightly oscillating. The intense parallel breathing solitons may also ride on the weak background which consists of central single breathing soliton and outer bound breathing soliton, as shown in Fig. 4(q1). In Fig. 4(a2), the bound breathing soliton and central single breathing soliton are sandwiched between the two intense parallel breathing solitons. And the former two kinds of solitons are relatively weak and have nearly the same intensities. The composite solitons in Fig. 4(i2) consist of three parts. In comparison with the case in Fig. 4(a2), the outer part is also the parallel breathing solitons which remains nearly unchanged only with its temporal trajectory oscillating very slightly. It is relatively intense. The second part can be referred to as the quasi-parallel breathing solitons for the reason that it has zigzag temporal trajectories. While the central part can be referred to as the chaotic bound breathing solitons because of its varied breathing period and temporal widths caused chaotic breathing behavior. In Figs. 4(b2), 4(e2), 4(h2), 4(k2), and 4(l2), the composite solitons consist of the outer quasi-parallel breathing solitons with zigzag temporal trajectories, the inner bound breathing solitons, and the central single breathing solitons. The three solitons have nearly equal intensities. Thus, in comparison with the case in Fig. 4(a2), the interaction between the outer quasi-parallel breathing solitons and the inner bound breathing solitons is more intense, which correspondingly results in the zigzag shifting and the outward moving respectively for the outer parallel solitons and the inner bound soliton in terms of their temporal trajectories. Besides, it is worth mentioning that, during the process of forming the corresponding solitons, some parts of energy will be radiated away in the form of dispersive waves. Moreover, we note that some dispersive waves are even as strong as the main solitons. For the reason that they keep nearly unchanged in intensities and widths when they depart far from the center in straight lines in the opposite directions, we may reasonably call them repulsive solitons. Furthermore, one may note that, in comparison with the case of N = 1 in Fig. 1, the soliton pulses for N > 1 generally have much more narrow pulse widths and shorter breathing periods.
By numerically calculating the evolutions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order in optical fibers, we reveal diverse breathing solitons. When the soliton order equals to 1, in the case of initial in-phase Airy pulses, we reveal single breathing solitons, bound breathing solitons, and single parallel breathing solitons. While in the case of initial out-of-phase Airy pulses, we observe parallel or repulsive soliton pairs with breathing or weak breathing. If the initial relative phases are arbitrary, the generated solitons may be moving single breathing solitons, parallel breathing solitons, or repulsive breathing soliton pairs. However, in comparison with the out-of-phase case, the repulsive breathing solitons pairs here consist of two asymmetrical breathing solitons which move in different direction and different velocities and have different breathing periods. Similarly, the two breathing solitons in the parallel breathing solitons here are also asymmetrical and different in terms of their breathing periods and pulse intensities. The most interesting and meaningful thing is the interactions of two in-phase truncated Airy pulses in case of N≠1. In this case, besides the ordinary single breathing solitons, bound breathing solitons, and parallel breathing solitons as discovered in case of N = 1, we surprisingly observe diverse composite solitons which consist of two or more than two kinds of breathing solitons. These composite solitons include double parallel breathing solitons, double bound breathing solitons, combination of weak bound breathing solitons inside and intense parallel breathing solitons outside, intense single parallel breathing solitons outside accompanied by weak bound breathing solitons and central single breathing solitons inside, combination of outer intense double parallel breathing solitons and inner weak bound breathing solitons, outer intense parallel breathing solitons accompanied with inner weak quai-parallel breathing solitons and chaotic bound breathing solitons, and outer quasi-parallel breathing solitons accompanied with the inner bound breathing solitons and the central single breathing solitons. This work enriches the types of solitons considerably and reveals more optical solitons for future experimental observation. What is more, for the reason that what kind of soliton can form eventually depends on the soliton order, initial pulse intervals, and initial relative phases, one can effectively manipulate and select the soliton expected and its evolution dynamics by adjusting these parameters. These special types of solitons may be potentially applied in designing special fiber devices which output optical pulses with special temporal profiles. They may expand the contents of future optical soliton communication by exploiting the composite solitons besides the conventional fundamental solitons. The shorter sub-pulses obtained in case of N > 1 may be extracted and applied in ultrafast imaging and other diverse ultrafast processes. Moreover, owing to the similarity of the temporal dispersion and spatial diffraction, these special temporal solitons here can also find their corresponding spatial counterparts and can be applied in optical signal processing.
Major Project of Natural Science Supported by the Educational Department of Sichuan Province in China Grants 13ZA0081 and 17ZA0072.
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