# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion
### Nonlinear Guidance
#### 2.1.1 Kerr Medium  
Initially, we consider the beam interaction in a Kerr NL medium, with \(\delta n=|\psi(x)|^{2}\). Since the energy is mainly stored in the main lobe of the Airy beam, a large distance between components will lead to a weak interaction, so we just consider the interaction for relatively small distances. The results are shown in Figure 2.  
Obviously, the major difference between the first two rows is the attraction of beams when the beams are in-phase and the repulsion when they are out-of-phase. Also visible is the breathing or the filamentation of the beams when they strongly interact. In the in-phase case, for a large distance, the two Airy components form two parallel solitons, as depicted in Figure 2a1,h1. With decreasing distance between the beams, the attraction between components increases, and the bound breathing solitons form. In general, the smaller the distance, the stronger the attraction and the smaller the period of soliton breathing. Curiously, the intensity image shown in Figure 2e1 has a smaller period than the one in Figure 2d1, even though in that case \(B=0\). The reason is that the main lobe of the Airy beam with \(B=0\) is located at about \(-1\), and there is still an interval between the two main lobes in the incidence, so the attraction is the biggest when \(B=1\) and the period of the formed soliton is then the smallest. The results for the out-of-phase beams are shown in Figure 2a2-h2. One can see that the soliton pairs formed from the incidence actually repel each other; the smaller the distance, the stronger the repulsion, until the beams overlap. Considering that the two Airy components are out-of phase, the main lobes will balance each other out at \(B=1\), so the soliton pair shown in Figure 2e2 is generated from the secondary lobes, while the other two come from main lobes. This is why the repulsion of  
Figure 2: Soliton formation in the interaction of two in-phase (**a1**–**h1**) and out-of-phase (**a2**–**h2**) incident Airy beams with \(A_{1}=A_{2}=3\), in the Kerr medium. (**a3**–**h3**) The same as (**a1**–**h1**), but with \(A_{1}=A_{2}=4\). The distance between beams is chosen by varying \(B\) (on the top of the figure). Black solid and dashed curves represent the ideal accelerating trajectories of the main lobes. Reproduced with permission from [35], Copyright the Optical Society of America, 2014.  
the soliton pair in Figure 2d2 is stronger than that in Figure 2e2. Notably, only two soliton pairs in Figure 2h2 are visible: the outer pair comes from the main lobes of the Airy components and the other from the secondary lobes. These results will be different when \(A\) is varied; for small \(A\) (less than 1), there will be no solitons generated; for large \(A\) (\(\sim\)10), multiple soliton pairs will be produced, but the propagation may become unstable because of the catastrophic self-focusing effect. The results for the in-phase beams when \(A_{1}=A_{2}=4\) are shown in Figure 2a3-h3; from these, one finds repulsion between the two solitons, especially in the cases \(B=0\) and \(B=1\). As shown in Figure 2e3, when \(B=1\), the refractive index change will make the solitons attract each other, and the attraction is quite strong over a long distance, but eventually the repulsion overtakes the attraction. The intensity of the superposed main lobes is enhanced, while the width is suppressed, in comparison with the case \(B=0\). Thus, the two solitons generated in the splitting of the overlapping main lobes will experience a smaller repulsion force than in Figure 2d3. When \(B\) is further increased, the main lobe of one component will superpose with the high-order lobes of the other component, so the solitons will come from the overlapping main and high-order lobes, as shown in Figure 2h3. When the distance between two solitons is large, their interaction becomes weak, and they propagate in parallel, as in Figure 2a,h.