# Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Numerical Results

#### 3.1. The Model with Diffraction Management

#### 3.2. The Model with Nonlinearity Management

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

LI | Lévy index |

FSE | Fractional Schrödinger equation |

NLSE | Nonlinear Schrödinger equation |

## References

- Berry, M.V.; Balazs, N.L. Diffraction-free beams. Am. J. Phys.
**1979**, 47, 264–267. [Google Scholar] [CrossRef] - Voloch-Bloch, N.; Lereah, Y.; Lilach, Y.; Gover, A.; Arie, A. Generation of electron Airy beams. Nature
**2013**, 494, 331–335. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Efremidis, N.K.; Chen, Z.G.; Segev, M.; Christodoulides, D.N. Airy beams and accelerating waves: An overview of recent advances. Optica
**2019**, 6, 686–701. [Google Scholar] [CrossRef] [Green Version] - Minovich, A.E.; Klein, A.E.; Neshev, D.N.; Pertsch, T.; Kivshar, Y.S.; Christodoulides, D.N. Airy plasmons: Non-diffracting optical surface waves. Laser. Phot. Res.
**2014**, 8, 221–232. [Google Scholar] [CrossRef] - Durnin, J. Exact solutions for nondiffracting beams. I. The scalar theory. J. Opt. Soc. Am. A
**1987**, 4, 651–654. [Google Scholar] [CrossRef] - Durnin, J.; Miceli, J.J.; Eberly, J.H. Diffraction-free beams. Phys. Rev. Lett.
**1987**, 58, 1499–1501. [Google Scholar] [CrossRef] - Gutiérrez-Vega, J.C.; Iturbe-Castillo, M.D.; Chávez-Cerda, S. Alternative formulation for invariant optical fields: Mathieu beams. Opt. Lett.
**2000**, 25, 1493–1495. [Google Scholar] [CrossRef] - Bandres, M.A.; Gutiérrez-Vega, J.C.; Chávez-Cerda, S. Parabolic nondiffracting optical wave fields. Opt. Lett.
**2004**, 29, 44–46. [Google Scholar] [CrossRef] - Recami, E.; Zamboni-Rached, M. Localized waves: A review. Adv. Imaging Electron. Phys.
**2009**, 56, 235–353. [Google Scholar] - Siviloglou, G.A.; Broky, J.; Dogariu, A.; Christodoulides, D.N. Observation of accelerating Airy beams. Phys. Rev. Lett.
**2007**, 99, 213901. [Google Scholar] [CrossRef] - Siviloglou, G.A.; Christodoulides, D.N. Accelerating finite energy Airy beams. Opt. Lett.
**2007**, 32, 979–981. [Google Scholar] [CrossRef] - Efremidis, N.K.; Christodoulides, D.N. Abruptly autofocusing waves. Opt. Lett.
**2010**, 35, 4045–4047. [Google Scholar] [CrossRef] [Green Version] - Papazoglou, D.G.; Efremidis, N.K.; Christodoulides, D.N.; Tzortzakis, S. Observation of abruptly autofocusing waves. Opt. Lett.
**2011**, 36, 1842–1844. [Google Scholar] [CrossRef] [Green Version] - Zhang, P.; Prakash, J.; Zhang, Z.; Mills, M.S.; Efremidis, N.K.; Christodoulides, D.N.; Chen, Z.G. Trapping and guiding microparticles with morphing autofocusing Airy beams. Opt. Lett.
**2011**, 36, 2883–2885. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chremmos, I.; Zhang, P.; Prakash, J.; Efremidis, N.K.; Christodoulides, D.N.; Chen, Z.G. Fourier-space generation of abruptly autofocusing beams and optical bottle beams. Opt. Lett.
**2011**, 36, 3675–3677. [Google Scholar] [CrossRef] [Green Version] - Liu, S.; Wang, M.R.; Li, P.; Zhang, P.; Zhao, J.L. Abrupt polarization transition of vector autofocusing Airy beams. Opt. Lett.
**2013**, 38, 2416–2418. [Google Scholar] [CrossRef] - Jiang, Y.F.; Huang, K.K.; Lu, X.H. Propagation dynamics of abruptly autofocusing Airy beams with optical vortices. Opt. Express
**2012**, 20, 18579–18584. [Google Scholar] [CrossRef] [PubMed] - Chen, B.; Chen, C.D.; Peng, X.; Peng, Y.L.; Zhou, M.L.; Deng, D.M. Propagation of sharply autofocused ring Airy Gaussian vortex beams. Opt. Express
**2015**, 23, 19288–19298. [Google Scholar] [CrossRef] [PubMed] - Li, P.; Liu, S.; Peng, T.; Xie, G.F.; Gan, X.T.; Zhao, J.L. Spiral autofocusing Airy beams carrying power-exponent-phase vortices. Opt. Express
**2014**, 22, 7598–7606. [Google Scholar] [CrossRef] - Polynkin, P.; Kolesik, M.; Moloney, J.V.; Siviloglou, G.A.; Christodoulides, D.N. Curved plasma channel generation using ultraintense Airy beams. Science
**2009**, 324, 229–232. [Google Scholar] [CrossRef] - Polynkin, P.; Kolesik, M.; Moloney, J.V. Filamentation of femtosecond laser Airy beams in water. Phys. Rev. Lett.
**2009**, 103, 123902. [Google Scholar] [CrossRef] [PubMed] - Clerici, M.; Hu, Y.; Lassonde, P.; Milián, C.; Couairon, A.; Christodoulides, D.N.; Chen, Z.G.; Razzari, L.; Vidal, F.; Légaré, F.; et al. Laser-assisted guiding of electric discharges around objects. Sci. Adv.
**2015**, 1, e1400111. [Google Scholar] [CrossRef] [Green Version] - Vettenburg, T.; Dalgarno, H.I.C.; Nylk, J.; Coll-Lladó, C.; Ferrier, D.E.K.; Cizmar, T.; Gunn-Moore, F.J.; Dholakia, K. Light-sheet microscopy using an Airy beam. Nat. Methods
**2014**, 11, 541–544. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nylk, J.; McCluskey, K.; Preciado, M.A.; Mazilu, M.; Yang, Z.; Gunn-Moore, F.J.; Aggarwal, S.; Tello, J.A.; Ferrier, D.E.; Dholakia, K. Light sheet microscopy with attenuation-compensated propagation-invariant beams. Sci. Adv.
**2018**, 4, eaar4817. [Google Scholar] [CrossRef] [Green Version] - Preciado, M.A.; Dholakia, K.; Mazilu, M. Generation of attenuation compensating Airy beams. Opt. Lett.
**2014**, 39, 4950–4953. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Baumgartl, J.; Mazilu, M.; Dholakia, K. Optically mediated particle clearing using Airy wavepackets. Nat. Photonics
**2008**, 2, 675–678. [Google Scholar] [CrossRef] - Cheng, H.; Zang, W.P.; Zhou, W.Y.; Tian, J.G. Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam. Opt. Express
**2010**, 18, 20384–20394. [Google Scholar] [CrossRef] - Zheng, Z.; Zhang, B.F.; Chen, H.; Ding, J.P.; Wang, H.T. Optical trapping with focused Airy beams. Appl. Opt.
**2011**, 50, 43–49. [Google Scholar] [CrossRef] - Salandrino, A.; Christodoulides, D.N. Airy plasmon: A nondiffracting surface wave. Opt. Lett.
**2010**, 35, 2082–2084. [Google Scholar] [CrossRef] - Minovich, A.; Klein, A.E.; Janunts, N.; Pertsch, T.; Neshev, D.N.; Kivshar, Y.S. Generation and near-field imaging of Airy surface plasmons. Phys. Rev. Lett.
**2011**, 107, 116802. [Google Scholar] [CrossRef] [Green Version] - Li, L.; Li, T.; Wang, S.M.; Zhang, C.; Zhu, S.N. Plasmonic Airy beam generated by in-plane diffraction. Phys. Rev. Lett.
**2011**, 107, 126804. [Google Scholar] [CrossRef] [PubMed] - Kaminer, I.; Segev, M.; Christodoulides, D.N. Self-accelerating self-trapped optical beams. Phys. Rev. Lett.
**2011**, 106, 213903. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bekenstein, R.; Segev, M. Self-accelerating optical beams in highly nonlocal nonlinear media. Opt. Express
**2011**, 19, 23706–23715. [Google Scholar] [CrossRef] [PubMed] - Fattal, Y.; Rudnick, A.; Marom, D.M. Soliton shedding from Airy pulses in Kerr media. Opt. Express
**2011**, 19, 17298–17307. [Google Scholar] [CrossRef] [PubMed] - Shen, M.; Gao, J.S.; Ge, L.J. Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media. Sci. Rep.
**2015**, 5, 9814. [Google Scholar] [CrossRef] [Green Version] - Hu, Y.; Sun, Z.; Bongiovanni, D.; Song, D.H.; Lou, C.B.; Xu, J.J.; Chen, Z.G.; Morandotti, R. Reshaping the trajectory and spectrum of nonlinear Airy beams. Opt. Lett.
**2012**, 37, 3201–3203. [Google Scholar] [CrossRef] [Green Version] - Hu, Y.; Li, M.; Bongiovanni, D.; Clerici, M.; Yao, J.P.; Chen, Z.G.; Azaña, J.; Morandotti, R. Spectrum to distance mapping via nonlinear Airy pulses. Opt. Lett.
**2013**, 38, 380–382. [Google Scholar] [CrossRef] [Green Version] - Malomed, B.A. Self-accelerating solitons. EPL
**2022**, 140, 22001. [Google Scholar] [CrossRef] - Dolev, I.; Kaminer, I.; Shapira, A.; Segev, M.; Arie, A. Experimental observation of self-accelerating beams in quadratic nonlinear media. Phys. Rev. Lett.
**2012**, 108, 113903. [Google Scholar] [CrossRef] - Mayteevarunyoo, T.; Malomed, B.A. Generation of χ
^{2}solitons from the Airy wave through the parametric instability. Opt. Lett.**2015**, 40, 4947–4950. [Google Scholar] [CrossRef] - Mayteevarunyoo, T.; Malomed, B.A. Two-dimensional χ
^{2}solitons generated by the downconversion of Airy waves. Opt. Lett.**2016**, 41, 2919–2922. [Google Scholar] [CrossRef] - Mayteevarunyoo, T.; Malomed, B.A. The interaction of Airy waves and solitons in the three-wave system. J. Optics.
**2017**, 19, 085501. [Google Scholar] [CrossRef] [Green Version] - Prasatsap, U.; Mayteevarunyoo, T.; Malomed, B.A. Two-dimensional Airy waves and three-wave solitons in quadratic media. J. Optics.
**2022**, 24, 055501. [Google Scholar] [CrossRef] - Liu, J.F.; Zhang, Y.Q.; Zhong, H.; Zhang, J.W.; Wang, R.; Belić, M.R.; Zhang, Y.P. Optical Bloch oscillations of a dual Airy beam. Ann. Phys.
**2017**, 530, 1700307. [Google Scholar] [CrossRef] - Wang, X.N.; Fu, X.Q.; Huang, X.W.; Yang, Y.J.; Bai, Y.F. The robustness of truncated Airy beam in PT Gaussian potentials media. Opt. Commun
**2018**, 410, 717–722. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2020**, 339, 1–77. [Google Scholar] [CrossRef] - Zaslavsky, G.M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep.
**2002**, 371, 461–580. [Google Scholar] [CrossRef] - Chechkin, A.C.; Gorenflo, R.; Sokolov, I.M. Fractional diffusion in inhomogeneous media. J. Phys. A Math. Gen.
**2005**, 38, L679–L684. [Google Scholar] [CrossRef] - Laughlin, R.B. Anomalous quantum hall efect: An incompressible quantum fuid with fractionally charged excitations. Phys. Rev. Lett.
**1983**, 50, 1395–1398. [Google Scholar] [CrossRef] [Green Version] - Rokhinson, L.P.; Liu, X.Y.; Furdyna, J.K. The fractional A.C. Josephson efect in a semiconductor-superconductor nanowire as a signature of Majorana particles. Nat. Phys.
**2012**, 8, 795–799. [Google Scholar] [CrossRef] [Green Version] - Momani, S.; Arqub, O.A.; Maayah, B.; Yousef, F.; Alsaedi, A. A reliable algorithm for solving linear and nonlinear Schrödinger equations. Appl. Comput. Math.
**2018**, 17, 151–160. [Google Scholar] - Sulem, C.; Sulem, P. The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse; Springer Series in Mathematical Sciences; Springer: Berlin, Germany, 1999. [Google Scholar]
- Bao, W.; Cai, Y. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Models
**2013**, 6, 1–135. [Google Scholar] [CrossRef] - Laskin, N. Fractional quantum mechanics. Phys. Rev. E
**2000**, 62, 3135–3145. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Laskin, N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A
**2000**, 268, 298–305. [Google Scholar] [CrossRef] [Green Version] - Laskin, N. Fractional Schrödinger equation. Phys. Rev. E
**2002**, 66, 056108. [Google Scholar] [CrossRef] [Green Version] - Longhi, S. Fractional Schrödinger equation in optics. Opt. Lett.
**2015**, 40, 1117–1120. [Google Scholar] [CrossRef] [PubMed] - Liu, S.; Zhang, Y.; Malomed, B.A.; Karimi, E. Experimental realisations of the fractional Schrödinger equation in the temporal domain. arXiv
**2022**, arXiv:2208.01128. [Google Scholar] - Huang, X.W.; Deng, Z.X.; Fu, X.Q. Dynamics of finite energy Airy beams modeled by the fractional Schrödinger equation with a linear potential. J. Opt. Soc. Am. B
**2017**, 34, 976–982. [Google Scholar] [CrossRef] - Huang, X.W.; Deng, Z.X.; Shi, X.H.; Fu, X.Q. Propagation characteristics of ring Airy beams modeled by fractional Schrödinger equation. J. Opt. Soc. Am. B
**2017**, 34, 2190–2197. [Google Scholar] [CrossRef] - Huang, X.W.; Shi, X.H.; Deng, Z.X.; Bai, Y.F.; Fu, X.Q. Potential barrier-induced dynamics of finite energy Airy beams in fractional Schrödinger equation. Opt. Express
**2017**, 25, 32560–32569. [Google Scholar] [CrossRef] - He, S.L.; Malomed, B.A.; Mihalache, D.; Peng, X.; Yu, X.; He, Y.J.; Deng, D.M. Propagation dynamics of abruptly autofocusing circular Airy Gaussianvortex beams in the fractional Schrödinger equation. Chaos Solitons Fractals
**2021**, 142, 110470. [Google Scholar] [CrossRef] - He, S.L.; Malomed, B.A.; Mihalache, D.; Peng, X.; He, Y.J. Propagation dynamics of radially polarized symmetric Airy beams in the fractional Schrödinger equation. Phys. Lett. A
**2021**, 404, 127403. [Google Scholar] [CrossRef] - Zhang, L.F.; Zhang, X.; Wu, H.Z.; Li, C.X.; Pierangeli, D.; Gao, Y.X.; Fan, D.Y. Anomalous interaction of Airy beams in the fractional nonlinear Schrödinger equation. Opt. Express
**2019**, 27, 27936–27945. [Google Scholar] [CrossRef] [PubMed] - He, S.L.; Zhou, K.Z.; Malomed, B.A.; Mihalache, D.; Zhang, L.P.; Tu, J.L.; Wu, Y.; Zhao, J.J.; Peng, X.; He, Y.J.; et al. Airy-Gaussian vortex beams in the fractional nonlinear Schrödinger medium. J. Opt. Soc. Am. B
**2021**, 38, 3230–3236. [Google Scholar] [CrossRef] - Malomed, B.A. Optical solitons and vortices in fractional media: A mini-review of recent results. Photonics
**2021**, 8, 353. [Google Scholar] [CrossRef] - Malomed, B.A. Soliton Management in Periodic Systems; Springer: New York, NY, USA, 2006. [Google Scholar]
- Eisenberg, H.S.; Silberberg, Y.; Morandotti, R.; Aitchinson, J.S. Diffraction management. Phys. Rev. Lett.
**2000**, 85, 1863–1866. [Google Scholar] [CrossRef] - Firth, W.J.; Skryabin, D.V. Optical solitons carrying orbital angular momentum. Phys. Rev. Lett.
**1997**, 79, 2450–2453. [Google Scholar] [CrossRef] - Malomed, B.A. Multidimensional Solitons; AIP Publishing: Melville, NY, USA, 2022. [Google Scholar]
- Wang, Q.; Zhang, L.; Malomed, B.A.; Mihalache, D.; Zeng, L. Transformation of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation by means of Lévy-index management. Chaos Solitons Fractals
**2022**, 157, 111995. [Google Scholar] [CrossRef] - Samko, G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach: New York, NY, USA, 1993. [Google Scholar]
- Duo, S.W.; Zhang, Y.Z. Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation. Comput. Math. Appl.
**2016**, 71, 2257–2271. [Google Scholar] [CrossRef] [Green Version] - Coutaz, J.L.; Kull, M. Saturation of the nonlinear index of refraction in semiconductor-doped glass. J. Opt. Soc. Am. B
**1991**, 8, 95–98. [Google Scholar] [CrossRef] - Tikhonenko, V.; Christou, J.; Luther-Davies, B. Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium. Phys. Rev. Lett.
**1996**, 76, 2698–2701. [Google Scholar] [CrossRef] [PubMed] - Malomed, B.A.; Matera, F.; Settembre, M. Reduction of the jitter for return-to-zero signals. Opt. Commun.
**1997**, 143, 193–198. [Google Scholar] [CrossRef] - Towers, I.; Malomed, B.A. Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity. J. Opt. Soc. Am. B
**2002**, 19, 537–543. [Google Scholar] [CrossRef] - Abdullaev, F.K.; Caputo, J.G.; Kraenkel, R.A.; Malomed, B.A. Controlling collapse in Bose-Einstein condensation by temporal modulation of the scattering length. Phys. Rev. A
**2003**, 67, 013605. [Google Scholar] [CrossRef] [Green Version] - Saito, H.; Ueda, M. Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate. Phys. Rev. Lett.
**2003**, 90, 040403. [Google Scholar] [CrossRef] [Green Version] - Itin, A.; Morishita, T.; Watanabe, S. Reexamination of dynamical stabilization of matter-wave solitons. Phys. Rev. A
**2006**, 74, 033613. [Google Scholar] [CrossRef]

**Figure 1.**The propagation dynamics of the ring-Airy beams under the combined action of the self-focusing saturable nonlinearity, defined by Equation (3) with values of ${\sigma}_{0}$ indicated in the top panels, and the periodic modulation of the diffraction coefficient, defined by Equation (8) with ${D}_{0}=1$ and $\Omega =2$, in the framework of Equation (6) with $\alpha =1.5$. The propagation is initiated by input (13) with parameters (15). (

**a1**,

**b1**,

**c1**): Side views (i.e., the cross-section drawn through $\eta =0$) of the propagating ring-Airy beams for different values of nonlinearity parameter ${\sigma}_{0}$ as indicated in the panels. Panels (

**a2**,

**b2**,

**c2**) show the amplitude of the optical field as the function of the propagation distance. Panels (

**a3**,

**b3**,

**c3**) present power spectra of the amplitude from (

**a2**,

**b2**,

**c2**): ${k}_{\zeta}$ is the respective wavenumber of the Fourier transform applied to the functions of $\zeta $.

**Figure 2.**The same as Figure 1 but for the case of self-defocusing saturable nonlinearity defined by Equation (6) with negative values ${\sigma}_{0}$ indicated in the panels.(

**a1**,

**b1**,

**c1**): Side views (i.e., the cross-section drawn through $\eta =0$) of the propagating ring-Airy beams for different values of nonlinearity parameter ${\sigma}_{0}$ as indicated in the panels. Panels (

**a2**,

**b2**,

**c2**) show the amplitude of the optical field as the function of the propagation distance. Panels (

**a3**,

**b3**,

**c3**) present power spectra of the amplitude from (

**a2**,

**b2**,

**c2**): ${k}_{\zeta}$ is the respective wavenumber of the Fourier transform applied to the functions of $\zeta $.

**Figure 3.**The propagation dynamics of ring-Airy beams under the action of the modulation format (9), as produced by simulations of Equation (6). Panels (

**a1**,

**b1**,

**c1**) display, in the cross-section $\eta =0$, the evolution of the propagating beams with the self-focusing nonlinearity, ${\sigma}_{0}=+1$, and LI values $\alpha =1.0$, $\alpha =1.4$, and $\alpha =1.6$, as indicated in the panels. The respective evolution of the amplitude of the optical field is displayed in panels (

**a2**,

**b2**,

**c2**).

**Figure 5.**The simulated propagation of the ring-Airy beams under the action of diffraction management with modulation format (8), where $\Omega =2$ is set again in combination with nonlinearity management (17) with $\tilde{\Omega}=4$, as produced by simulations of Equation (16). Panels (

**a1**,

**b1**,

**c1**) and (

**a2**,

**b2**,

**c2**) display, in the cross-section $\eta =0$, the propagation dynamics of the propagating beams with a fixed value of $\alpha =1.5$ for different values of the self-focusing and defocusing nonlinearity parameter ${\sigma}_{0}$, as indicated in the panels. Panels (

**a3**,

**b3**,

**c3**) display the results for fixed ${\sigma}_{0}=0$ and different values of the LI: $\alpha =1.0$, $\alpha =1.5$, and $\alpha =1.9$, respectively.

**Figure 6.**The propagation dynamics of the ring-Airy beams under the combined action of the decaying diffraction-modulation format (9) and nonlinearity management (17) with ${\sigma}_{0}=1$ and $\tilde{\Omega}=20$, as produced by simulations of Equation (16). Panels (

**a1**,

**b1**,

**c1**) display, in cross-section $\eta =0$, the evolution of the beams with fixed values of $\alpha $, as indicated in the panels. Panels (

**a2**,

**b2**,

**c2**) show the corresponding evolution of the amplitude of the optical field.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, P.; Wei, Y.; Malomed, B.A.; Mihalache, D.
Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations. *Symmetry* **2022**, *14*, 2664.
https://doi.org/10.3390/sym14122664

**AMA Style**

Li P, Wei Y, Malomed BA, Mihalache D.
Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations. *Symmetry*. 2022; 14(12):2664.
https://doi.org/10.3390/sym14122664

**Chicago/Turabian Style**

Li, Pengfei, Yanzhu Wei, Boris A. Malomed, and Dumitru Mihalache.
2022. "Stabilization of Axisymmetric Airy Beams by Means of Diffraction and Nonlinearity Management in Two-Dimensional Fractional Nonlinear Schrödinger Equations" *Symmetry* 14, no. 12: 2664.
https://doi.org/10.3390/sym14122664