Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media

We investigate the spatially optical solitons shedding from Airy beams and anomalous interactions of Airy beams in nonlocal nonlinear media by means of direct numerical simulations. Numerical results show that nonlocality has profound effects on the propagation dynamics of the solitons shedding from the Airy beam. It is also shown that the strong nonlocality can support periodic intensity distribution of Airy beams with opposite bending directions. Nonlocality also provides a long-range attractive force between Airy beams, leading to the formation of stable bound states of both in-phase and out-of-phase breathing Airy solitons which always repel in local media.

However, all the works mentioned above were investigated in local nonlinear media. It has been shown that the boundary conditions of a strongly nonlocal media affect deeply the propagation dynamics of self-accelerating beams 35 . Recently, an analytical expression of an Airy beam propagating in a strongly nonlocal nonlinear media was derived to show the normalized intensity distribution of the Airy beam is always periodic 36 . In optical domain, generally, nonlocal nonlinearity means that the light-induced refractive index change of a material at a particular location is determined by the light intensity in a certain neighborhood of this location. Such a nonlocal optical nonlinearity exists in nematic liquid crystals 37 and thermal media 38 . Many works have shown that nonlocality has profound effects on the solitons propagation 39 . Nonlocal nonlinearity also affects the interaction 38 of out-of-phase bright solitons [40][41][42] and dark solitons [43][44][45] .
In this paper, we investigate the solitons shedding from Airy beams and anomalous interactions of Airy beams in nonlocal nonlinear media numerically. We find that nonlocality has profound effects on the propagation dynamics of the solitons shedding from Airy beam. It is also shown that the strong nonlocality can support periodic intensity distribution of Airy beams with opposite bending directions. Nonlocality also provides a longrange attractive force between Airy beams, leading to the formation of stable bound states of both in-phase and out-of-phase breathing Airy solitons which always repel in local media.

Results
Dynamics of solitons shedding from airy beams in nonlocal nonlinear media. We consider an Airy beam propagating in a medium with a self-focusing nonlocal cubic nonlinearity. The envelope y(x,z) of the Airy beam is described by the normalized nonlocal nonlinear Schrodinger equation, where R(x) corresponds to the normalized nonlocal response function. Without loss of generality, we consider the case of so-called Gaussian nonlocal response functions 46 : , with the characteristic width s to represent the degree of nonlocality. It describes a local and a strongly nonlocal media when s?0 and a s?? 39 , respectively. In general, the realistic forms of the nonlocal response functions depend on the underlying physical process of the materials 47 . It has been shown that as long as the response function is monotonically decaying, the physical properties of solitons in nonlocal media do not depend strongly on the shape of the nonlocal response function 48 . Although the Gaussian nonlocal response function is phenomenological, it can describe the general properties of other actual nonlocal media 47 . Firstly, we study the propagation of a finite power Airy beam in nonlocal nonlinear media by considering the exponentially decaying version 6 where A is the amplitude of the Airy beam, and aw0 is the the decay factor to ensure containment of the infinite Airy tail and the finite power of the Airy beam 6 . For simplicity, we set a~0:2 throughout this paper. In Fig. (1), we show the propagation dynamics of Airy beams in nonlocal nonlinear media by direct numerical integration of Eq. (1) with split-step Fourier transform method. Firstly, we consider the case of local cubic media with s~0 [Figs. 1(a-c)]. For small amplitude (low power), as shown in Fig. 1(a), the Airy beam performs the acceleration in space and subsequently it succumbs to diffraction. However, when A is sufficiently large 28,30 , a stationary soliton will be formed out of the centered energy about the Airy main lobe [ Fig. 1 (c)]. The intensity distribution of the Airy beam (z~0) and the soliton (z~30) are displayed in Figs. 2(a,b). It is obvious that the peak intensity of the soliton is larger than that of the main lobe of the Airy beam, which indicates that the tail power of the Airy beam is almost confined into the main lobe to shed a soliton. The soliton exhibits periodic oscillations in the soliton amplitude and width. Similar phenomena has been studied previously 28 , which is not our aim in this paper.
If we keep the amplitude of the soliton [ Fig. 1(c)] invariant, the nonlocality (e.g., s~1:5) will weaken the stability of the soliton, as shown in Fig. 1(d). The period and intensity of oscillation in the soliton amplitude will become bigger and its width will become larger [ Fig. 1(d)]. Although nonlocality can provide a long range attractive force to stabilize complex solitons states, it will always weaken the strength of the nonlinearity 39 . A stable soliton always require a larger amplitude with the increases of s 46 . As shown in Figs. 1(d-f) (s~1:5), the stability of the Airy soliton is better improved when the amplitude increase, exhibited as the decrease of the beam width as well as the period and intensity of oscillation in the soliton amplitude. The intensity of the soliton keeps almost uniform for sufficiently large amplitude with snake oscillations [ Fig. 1(f)]. From Figs. 1(c, f), we can also see that when a soliton is formed, the soliton will always take the main part of the initial input power, while a small fraction of the power is transformed into a self-accelerating linear packet. This phenomena is similar with the solitons dynamics obtained by using the Zakharov-Shabat scattering problem in local media 30 .
It is also interesting that the nonlinear media cannot support stationary solitons but the periodic intensity distribution of Airy beam with opposite bending directions in strongly nonlocal regime, as shown in Fig. (3). The intensity of these beam maintain good Airylike profile with a main lobe and the decaying tails, as shown in Figs. 2 (c,d). Our numerical results of periodic intensity distribution of Airy beam obtained here agree with the analytical works of Airy beam in strongly nonlocal media very well 36 . The period will become larger when the degree of nonlocality increases or the amplitude decrease. The reason is that in the regime of strong nonlocality, the nonlocal nonlinearity trends to linear 39 which is hard to trap the Airy beam into a soliton. However, such a strong nonlocality have a strong impact on the beam trajectory 35 : it can cause the Airy beam bending in a opposite direction periodically 36 . Besides nonlocal nonlinear media 35,36 , the similar phenomenon of periodic intensity distribution of Airy beam with opposite bending directions also occurs to other different physical systems, such as in plasmons with a linear refractive index profile 49,50 and in curved space 51 .
Anomalous interactions and bound states of airy solitons in nonlocal media. Next, we focus on the anomalous interactions   and bound states of Airy solitons in nonlocal nonlinear media. We assume that the incident beam is composed of two shifted counter propagating Airy beams with a relative phase between them, where r is the parameter controlling the phase shift and B is the parameter controlling beam separations. In this report, we consider both in-phase and out-of-phase Airy beams with r~0 and r~1, respectively 33,34 .
Bound states of in-phase airy solitons. In Fig. (4), we show the interactions between in-phase Airy beams (r~0) with some different beam separations. For comparison, we also re-do some previous results in local media (s~0) with A~3 33,34 . For larger separations, the two Airy components form two parallel solitons 33 Fig. 4(d)]. The nonlocality provides a long range attractive force which can enhance the interaction between solitons 52 . This is true for larger separated Airy beams [ Fig. 4(f)]. For smaller separated in-phase Airy beams, the attractive force between them is big enough to form bound states with only local nonlinearity [ Fig. 4(a)]. However, nonlocality will always weaken the interactions of smaller separated in-phase Airy beams [ Fig. 4(d)] because of nonlocality can weaken the strength of the nonlinearity 39 .
Increasing the amplitude to A~4, in local media, compared with the Figs. 4(a-c), the interactions are weakened obviously [Figs. 5(a, b)] except for the larger separated beam [ Fig. 5(c)]. In particular, the repulsion appears for the case B~1, as shown in Fig. 5(a) 34 . The repulsion can be balanced by the attractive force induced even by a small degree of nonlocality. We show such bound state of breathing solitons in Fig. 5(d) 34 . For a given amplitude A~3, Figs. 6(d-f) clearly show the nonlocality (s~1) will weaken the interactions in spite of the separation is large or small due to the fact that the nonlocality cannot balance the repulsion of the out-of-phase beams.
Stationary bound states of out-of-phase Airy solitons may be obtained with larger amplitude in local media when the strong self-focusing effect balances the out-of-phase repulsion, as shown in Figs. 7(a-c). We consider the strongly repulsive case with B~0,1,2. From Figs. 7(a-c), we can find that Airy beams with smaller separation always need larger amplitude to form a stable soliton pairs, e.g., A~12 for B~0 [ Fig. 7(a)]. Interestingly, we can even obtain two soliton pairs with different intensities at small separation B~0 [ Fig. 7(a)]. The soliton pairs cannot obtained from in-phase Airy beams with larger amplitude due to that strong self-focusing and in-phase attractive force always break up the beam, leading to the collapse of the Airy beams.
The soliton pairs obtained in local media [Figs. 7(a-c)] with large amplitude may become unstable because of the strong self-focusing effect 34 . After some propagation distances, e.g., z~50,160,70 for B~0,1,2, the repulsion will overtake the attraction and soliton pairs will fly away in opposite directions 34 . Completely stable out-of-phase   . In fact, the stable bound states of breathing soliton pairs is a result of balance between the effects of nonlocal, nonlineaity, diffraction, and repulsion. Nonlocality provides a long range attractive force to balance the repulsion of out-of-phase Airy beams, leading to the formation of bound state which always repel in local media.

Discussions
One of the most important and interesting dynamic of the solitons is their particle-like interactions 53 . In purely local nonlinear media, the bright solitons may attract, repel, and even form bound states, depending on their relative phase 54 , whereas, the interaction of dark solitons is always repulsive 55 . Recently, the interactions of solitons have been investigated in media with spatially nonlocal nonlinearity. Nonlocal nonlinearity provides a long-range attractive force, leading to the formation of stable bound states of both out-of-phase bright solitons [40][41][42] and dark solitons [43][44][45] . This long-range nonlocal nonlinearity also allow people to observe experimentally the multipole solitons-arrays of out-of-phase bright spots 56 .
Up to now, the interaction dynamics of Airy beam have only been investigated in local media [31][32][33][34] . For out-of-phase Airy beams, they are always repel in local media 33,34 . This repulsive force also exist in the in-phase Airy beams when their amplitudes (energy) are large enough 34 . Thus, a question arise naturally: can one obtain stable bound states (soliton pairs) of Airy beams in nonlinear media?
In this report, with the help of nonlocality, we have obtained such bound states (soliton pairs) of in-phase as well as out-of-phase Airy beams in nonlocal nonlinear media. We also numerically check the stability of the bound states. We perturb the initial Airy beams by 5% random noise perturbations and then simulate their evolutions numerically. As shown in Fig. (8), we only consider the case of out-of-phase Airy beams. We can see that the bound states of breathing Airy solitons are surely stable and propagate robustly against perturbations. In summary, nonlocality provides a long-range attractive force on Airy beams, leading to the formation of stable bound states of breathing Airy solitons which always repel in local media.

Methods
Split-step fourier transform method. In our numerical simulations of the propagation dynamics of Airy beams and interactions of them, the split-step Fourier tansform method is used to integrate the nonlinear Schrödinger equation [Eq. (1)]. This method relies on computing the solution in small steps, and treating the linear and the nonlinear steps separately 57 . Firstly, we rewrite Eq. (1) as withD andN are the linear and nonlinear operator, N~i respectively. For an optical beam y(x, z) at propagation distance z, in the next step dz, the optical field distribution y(x, zzdz) can be obtained with the Split-step method.
In the first half of the step dz=2, we only consider the effect of the linear operator and then we consider the nonlinear operator in the whole step dz finally, in the second half of the step dz=2, the linear operator is considered again and the optical field distribution y(x,zzdz) can be obtained as y(x, zzdz)~expD dz 2 exp (Ndz) expD dz 2 y(x, z): Fourier transform of the nonlocal convolution term. In the numerical simulations with the split-step Fourier transform method, the Fourier transform of the convolution term of the nonlinear refractive index change dn(I)Ð R(x{x')jy(x', z)j 2 dx' should be addressed here