Self-accelerating Airy beams have become the focus of many theoretical and experimental investigations^[1,2] due to their possible applications in particle clearing,^[3] light bullets,^[4] curved plasma channel generation,^[5] optical routing,^[6] optical interconnects,^[7] image signal transmission,^[8] vacuum electron acceleration,^[9] abruptly autofocusing beams,^[10] and so on. In the last decade, besides their use in linear media, such beams have also been extensively researched in many nonlinear materials, such as Kerr nonlinear dielectrics,^[11,12] photorefractive media,^[13–16] nonlocal nonlinear media,^[17] Bose–Einstein condensates,^[18] metal surfaces,^[19] quadratic media,^[20] highly non-instantaneous cubic media,^[21] and photonic crystals.^[22] In particular, solitons can be generated from Airy beams because of the existence of nonlinearity,^[11–15] which propagate without transverse acceleration and perform a breathing state. The interactions between Airy beams can also result in the generation of single solitons or soliton pairs.^[17,23–26] More recently, such solitons have been observed experimentally in photorefractive media.^[13,15]

As a generalized form of Airy beams, Airy–Gaussian (AiG) beams provide a more realistic approach to the description of Airy beams, which carry finite power and retain non-diffracting properties within a finite propagation distance, and can achieve a very good approximation in experiment.^[27] So far, many researchers have theoretically and experimentally studied the propagation and interactions of AiG beams.^[28–37] For example, Deng et al. investigated the propagation and properties of AiG beams in a Kerr medium,^[28] quadratic-index medium,^[29] strongly nonlocal nonlinear media,^[30] and uniaxial crystals.^[31] They also discussed the interactions of AiG beams in Kerr and saturable media.^[32,33] Furthermore, Zhang reported the bound states of breathing AiG beams in a nonlocal nonlinear medium.^[34] Shi et al.^[35] discussed the interactions of AiG beams in photonic lattices with defects. Gao et al.^[36] analyzed the radiation force functions of two AiG beams on a cylindrical particle. We also studied in detail the propagation and interactions of AiG beams in saturable nonlinear media in a recent article.^[37]

However, due to strong focusing nonlinearity, unstable soliton propagation may occur in cubic media for intense excitation at the input plane.^[12] So, how can we obtain stable solitons during the propagation of AiG beams? How can we control the propagation of these breathing solitons? What factors can affect the propagation and interactions of AiG beams? These questions are very interesting.

In this paper, we introduce a higher-order nonlinearity, such as the defocusing quintic type, which can suppress the strong cubic nonlinearity so as to obtain stable breathing solitons. In one transverse dimension, we first numerically study the propagation of single AiG beams in a cubic–quintic nonlinear medium, and then the stability and the propagation control of the solitons produced by the AiG beams. In addition, we briefly investigate the interactions of two AiG beams by adjusting their initial amplitudes, phase shift, beam interval, and the initial light distribution factor.

Under paraxial approximation, we consider the generalized nonlinear Schrödinger equation in the normalized one transverse dimension form^[38]

Fig. 1.

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	Fig. 1. (color online) (a) Intensity distributions of AiG beams with g = 0.01, 0.1, and 1. (b) Propagation of AiG beam in free space for g = 0.01.

We have shown the single beam solutions in Eq. (2). In fact, to investigate the interactions of AiG beams, we should construct more complex incident beams, composed of two shifted counter-propagating single AiG beams with a relative phase shift and different amplitudes,^{[23–25,35,37]}

To investigate the propagation of a single AiG beam in a cubic–quintic nonlinear medium, we have implemented the well-known split-step Fourier transform methods to solve Eq. (1) by varying the initial amplitude A₀ and the field distribution factor g. The corresponding results are shown in Fig. 2. Firstly, we consider a typical case of pure Kerr nonlinearity with ε = 0 and g = 0.01. For a small initial amplitude A₀ (low power), the AiG beam accelerates in the transverse direction and eventually succumbs to diffraction in Fig. 2(a1). With an increase of A₀ in Fig. 2(a2), it is shown that a single breathing soliton can emerge from the centered energy around the AiG main lobe, and it moves along a straight line. Actually, this phenomenon has been reported in Refs. [12] and [17]. Unfortunately, when A₀ continues to increase, it is clear that the soliton width quickly narrows in Fig. 2(a3), and the soliton can even fail to form for intense excitation at the input plane in Fig. 2(a4) due to the strong focusing nonlinearity, which means that the formation of a breathing soliton should also impose an upper threshold on the initial amplitude.^[39]

Fig. 2.

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	Fig. 2. (color online) Soliton generation in the propagation of a single AiG beam in a cubic–quintic nonlinear medium with changing values of A0, g, and ε.

In this paper, our focus is on the case of ε = − 1 where the defocusing quintic nonlinearity will weaken the focusing cubic nonlinearity, so as to strengthen the stability of the breathing soliton. As seen in Figs. 2(b1)–2(b4), although a stable breathing soliton always requires a larger initial amplitude and no soliton exists for sufficiently high initial amplitude, a notable difference is that the soliton width can remain almost invariant because of the suppression of quintic nonlinearity. It is worth mentioning that the soliton intensity will be weakened for higher values of g in Figs. 2(c1)–2(c4). When g is large enough, a stable soliton needs a larger initial amplitude (see Figs. 2(d1)–2(d4)). That is to say, the formation of a soliton is easier when the AiG beam is close to the Airy distribution than when it is close to the Gaussian distribution.

We have examined the relations between the soliton oscillations and the incident AiG beam amplitude. One can see in Fig. 3(a) that the soliton breathing period decreases monotonically with the increase of A₀. Here, the soliton breathing period is defined as the distance between adjacent peaks or troughs of the oscillating soliton envelope. In Fig. 3(b), the oscillations of the soliton maximum intensity are shown as functions of the propagation distance for selected values of A₀ corresponding to the red circles in Fig. 3(a). As can be seen, the soliton exhibits damped oscillations in the soliton maximum intensity. The maximum intensity increases with increasing initial amplitude. Similar behavior in Kerr or saturable nonlinear media has also been reported in Refs. [12] and [37].

Fig. 3.

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	Fig. 3. (color online) (a) Breathing period of the soliton as a function of the initial amplitude A0 for selected light field distribution factor g = 0.01, 0.5, and 1. (b) Oscillations of the soliton maximum intensity with three different initial amplitudes A0 for g = 0.01.

To check the stability properties of these breathing solitons, we simulate their evolution under some random perturbations. As an example, figures 4(a1)—4(c1) depict the initial AiG intensities perturbed by 20% random noise perturbations for g = 0.01, 0.1, and 1, and the corresponding evolution simulation results are shown in Figs. 4(a2)—4(c2). It is evident that the solitons propagate robustly against perturbations (as far as 100 Rayleigh range), and hence they are certainly stable.

Fig. 4.

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	Fig. 4. (color online) Intensity evolution of AiG beams under 20% random noise perturbations with A0 = 3 for selected g = 0.01, 0.1, and 1. (a1)—(c1) The input AiG beam intensities. (a2)—(c2) The evolution with perturbation corresponding to panels (a1)—(c1), respectively.

In order to control the propagation direction of the breathing soliton, we introduce a launch angle to the incident AiG beam. For computer modeling of this process, we solved Eq. (1) with the initial condition that^[40,41]

In Figs. 5(a1)–5(d1), the dynamics of an AiG beam with different values of v is presented for given values of A₀ = 2 and g = 0.01. We plot the evolution of the AiG beam intensity in order to see the influence of the launch angle on the propagation of the single breathing soliton. The propagation direction of the soliton will tilt to the left when v ˂ 0 and to the right when v ˃ 0. The bigger the value of |v|, the greater the tilt angle of the soliton is. Therefore, the propagation direction of the breathing soliton can be adjusted by controlling the launch angle of the AiG beam. We also noticed that the change of launch angle has no effect on the intensity and width of the soliton. Figures 5(a2)–5(d2) show the soliton intensity profiles when they are propagated at ξ = 10 (corresponding to the position of the red dashed line in Figs. 5(a1)–5(d1)). As a result, except for the transverse position, the intensity and width of the soliton remain unchanged with the change of the launch angle.

Fig. 5.

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	Fig. 5. (color online) Intensity evolution of an AiG beam with different launch angles with A0 = 2, g = 0.01, v = (a1) − 2, (b1) −1, (c1) 1, and (d1) 2. (a2)–(d2) The intensity of the soliton at propagation distance ξ = 10 corresponding to the location of the red dashed line in panels (a1)–(d1), respectively.

In this section, we will focus on interactions of AiG beams in a cubic–quintic nonlinear medium. The split-step Fourier method is used to solve Eq. (1) with competing nonlinearity under the initial condition of Eq. (3).

Firstly, when A₁ = A₂ = 1.4 and g = 0.01, numerical simulation results with different phase shifts δϕ and beam intervals B (relatively small) are depicted in Fig. 6. Obviously, the in-phase AiG beams (δϕ = 0 in Figs. 6(a1)–6(e1)) attract each other and single solitons can form, while the out-of-phase AiG beams (δφ = π in Figs. 6(a3)–6(e3)) repel each other and soliton pairs can form. Similar behavior has been reported in Refs. [23] and [24]. It is interesting to note that in Figs. 6(a2)–6(e2), when δϕ = π/2, asymmetric soliton pairs are visible for selected values of B. If B ≤ 0, the intensity of the soliton on the right side is stronger than that on the left side. However, if B ˃ 0, the intensity of the soliton on the right side is weaker than that on the left side. With a decreasing interval |B|, the intensity of the weak side soliton becomes weaker and almost disappears. A contrary result appears in Figs. 6(a4)–6(e4) with the case of δϕ = 3π/2. Thus, we can control the interaction of AiG beams to obtain different breathing solitons or soliton pairs by adjusting the phase shift and interval between the two AiG beams.

Fig. 6.

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	Fig. 6. (color online) The interactions of two AiG beams with different phase shifts and beam intervals. Other parameters are A1 = A2 = 1.4 and g = 0.01.

Secondly, as an example, for B = 3 in the in-phase case, we consider the interaction of two AiG beams for different A₁ and A₂ in cubic–quintic nonlinear media, as shown in Fig. 7. When g = 0.01, the initial light fields of the two AiG beams approach the Airy distribution. After shedding some radiation from the main lobes of the two AiG beams initially, single breathing solitons can form in the center of the two launched AiG beams, and symmetric soliton pairs can also be generated on both sides of the beam center (Figs. 7(a1)–7(a4)). It is important to note that in Fig. 7(a1) a soliton pair cannot form for small initial amplitudes because of the weak nonlinear focusing effect. With increasing initial amplitudes, the focusing nonlinearity becomes stronger so that one soliton pair can form in Fig. 7(a2), and the intensity of the single soliton is higher than that of the soliton pair. With further increase in the initial amplitudes, we can see in Figs. 7(a3) and 7(a4) that the intensity of the soliton pair gradually becomes larger and the intensity of the single soliton becomes smaller. When g = 0.1, only single breathing solitons can form. The bigger the initial amplitudes are, the larger the soliton intensity is and the faster the soliton breathing rate is (Figs. 7(b1)–7(b4)). When g = 1, the initial light fields of the two AiG beams are close to the Gaussian distribution, the initial amplitudes at which stable single breathing solitons can be formed are higher than those when g is smaller (Figs. 7(c1)–7(c4)).

Fig. 7.

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	Fig. 7. (color online) The interactions of two AiG beams with different initial amplitudes and light field distribution factors. Other parameters are δ ϕ = 0 and B = 3.

We have numerically investigated the generation of breathing solitons in the propagation and interactions of AiG beams in a cubic–quintic nonlinear medium in one transverse dimension by using the split-step Fourier transform method. We find that stable breathing solitons can be produced in the propagation of single AiG beams due to the compensation of the defocusing quintic nonlinearity on the focusing cubic nonlinearity within a certain range of the initial amplitude A₀. The soliton breathing period decreases and the maximum intensity of the soliton increases monotonically with increasing A₀. The propagation direction of the solitons can be adjusted by controlling the launch angle of the AiG beams. In addition, numerical simulations of the interactions between two AiG beams are carried out in a cubic–quintic nonlinear medium.The results show that different single breathing solitons or soliton pairs can be obtained by adjusting the phase shift and interval between two AiG beams for given values of the initial amplitude. When the initial amplitudes are changed, the interactions of the two AiG beams can lead to single breathing solitons and symmetrical soliton pairs when the initial light field distribution tends to be the Airy distribution. The intensity of the symmetrical soliton pair increases and the intensity of the single breathing soliton decreases with increase of the initial amplitudes. When the initial light field distribution is close to the Gaussian distribution, only single breathing solitons can form. The bigger the initial amplitudes are, the larger the soliton intensity is and the faster the soliton breathing rate is.