[1] Laskin N. Fractional quantum mechanics and Lévy path integrals[J]. Physics Letters A, 268, 298-305(2000).

[2] Laskin N. Fractional Schrödinger equation[J]. Physical Review E, 66, 056108(2002).

[3] Longhi S. Fractional Schrödinger equation in optics[J]. Optics Letters, 40, 1117-1120(2015).

[4] Zhang Y Q, Liu X, Belić M R et al. Propagation dynamics of a light beam in a fractional Schrödinger equation[J]. Physical Review Letters, 115, 180403(2015).

[5] Zhang Y Q, Zhong H, Belić M R et al. Diffraction-free beams in fractional Schrödinger equation[J]. Scientific Reports, 6, 23645(2016).

[6] Zhang Y Q, Wang R, Zhong H et al. Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation[J]. Scientific Reports, 7, 17872(2017).

[7] Huang C M, Dong L W. Beam propagation management in a fractional Schrödinger equation[J]. Scientific Reports, 7, 5442(2017).

[8] Zang F, Wang Y, Li L. Dynamics of Gaussian beam modeled by fractional Schrödinger equation with a variable coefficient[J]. Optics Express, 26, 23740-23750(2018).

[9] Xin W, Song L J, Li L. Propagation of Gaussian beam based on two-dimensional fractional Schrödinger equation[J]. Optics Communications, 480, 126483(2021).

[10] Zhang Y G, Wu Z K, Ru J M et al. Evolution of the Bessel-Gaussian beam modeled by the fractional Schrödinger equation[J]. Journal of the Optical Society of America B, 37, 3414-3421(2020).

[11] Tan C, Liang Y, Zou M et al. Dynamics of quadratic phase controlled Hermite-Gaussian beams in fractional systems based on different variable coefficients and potentials[J]. Acta Physica Sinica, 73, 134205(2024).

[12] Chen W J, Lu K Q, Hui J L et al. Propagation and interactions of Airy-Gaussian beams in saturable nonliear medium[J]. Acta Physica Sinica, 65, 244201(2016).

[13] Xiao Y, Wang P X, Guo T et al. Propagation of dual Airy-Gaussian beams in Gaussian potential under fractional effect[J]. Chinese Journal of Lasers, 48, 2005002(2021).

[14] Luo J, Ji X L, Li X Q. Influence of nonlinear self-focusing on airy beams propagating upwards in the atmosphere[J]. Chinese Journal of Lasers, 51, 0505001(2024).

[15] Chen W J, Wang T, Wang J et al. Dynamics of interacting Airy beams in the fractional Schrödinger equation with a linear potential[J]. Optics Communications, 496, 127136(2021).

[16] Ren S M, Gao R, Guo T et al. Periodic evolution of the Pearcey Gaussian beam under fractional effect[J]. Journal of Physics B: Atomic Molecular Physics, 55, 205401(2022).

[17] Liu S L, Zhang Y W, Malomed B A et al. Experimental realisations of the fractional Schrödinger equation in the temporal domain[J]. Nature Communications, 14, 222(2023).

[18] Xiao J, Tian Z X, Huang C M et al. Surface gap solitons in a nonlinear fractional Schrödinger equation[J]. Optics Express, 26, 2650-2658(2018).

[19] Li L, Li H G, Ruan W et al. Gap solitons in parity-time-symmetric lattices with fractional-order diffraction[J]. Journal of the Optical Society of America B, 37, 488-494(2020).

[20] Zhang B J, Lu K Q, Zhao C et al. Research on defect solitons in Kagome photonic lattices in centrosymmetric photorefractive crystals[J]. Chinese Journal of Lasers, 43, 1208001(2016).

[21] Wang S Y, Chen W J, Liu W J et al. Two-dimensional line defect lattice solitons in nonlinear fractional Schrödinger equation[J]. Optics & Laser Technology, 175, 110870(2024).

[22] Rizvi S T R, Seadawy A R, Younis M et al. Various optical soliton for a weak fractional nonlinear Schrödinger equation with parabolic law[J]. Results in Physics, 23, 103998(2021).

[23] Wen J M, Bo W B, Wen X K et al. Multipole vector solitons in coupled nonlinear Schrödinger equation with saturable nonlinearity[J]. Acta Physica Sinica, 72, 100502(2023).

[24] Li S Q, Zhang X, Lin J. Propagation of dark soliton in nonlocal nonlinear coupler[J]. Acta Physica Sinica, 70, 184206(2021).

[25] Zhai Y B, Li R J, Li P F. Symmetry breaking of partially PT-symmetric solitons in fractional diffraction systems[J]. Acta Optica Sinica, 44, 0519002(2024).

[26] Zhang L F, Li C X, Zhong H Z et al. Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes[J]. Optics Express, 24, 14406-14418(2016).

[27] Wang T, Lian C, Wang L K et al. Interaction of Airy beams and soliton in saturable nonlinear medium[J]. Chinese Journal of Lasers, 48, 1308001(2021).

[28] Chen W J, Lian C, Luo Y A. Interaction of Airy beams modeled by the fractional nonlinear cubic-quintic Schrödinger equation[J]. Physica Scripta, 96, 125256(2021).

[29] He S L, Zhou K Z, Malomed B A et al. Airy-Gaussian vortex beams in the fractional nonlinear-Schrödinger medium[J]. Journal of the Optical Society of America B, 38, 3230-3236(2021).

[30] Siviloglou G A, Christodoulides D N. Accelerating finite energy Airy beams[J]. Optics Letters, 32, 979-981(2007).

[31] Broky J, Siviloglou G A, Dogariu A et al. Self-healing properties of optical Airy beams[J]. Optics Express, 16, 12880-12891(2008).

[32] Fattal Y, Rudnick A, Marom D M. Soliton shedding from Airy pulses in Kerr media[J]. Optics Express, 19, 17298-17307(2011).

[33] Zhang Y Q, Belić M R, Zheng H B et al. Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media[J]. Optics Express, 22, 7160-7171(2014).

[34] Bai X Q, Wang Y H, Zhang J et al. Soliton shedding from Airy-Gaussian beams in photorefractive media[J]. Chinese Journal of Lasers, 46, 0805001(2019).

[35] Chen W J, Ju Y, Liu C Y et al. Generation of breathing solitons in the propagation and interactions of Airy-Gaussian beams in a cubic-quintic nonlinear medium[J]. Chinese Physics B, 27, 114216(2018).

[36] Zhan K Y, Yang Z D, Jiao R Y et al. Controllable interaction of Airy beams via initial launch angle in Kerr media[J]. Optics Communications, 432, 49-53(2019).

[37] Aleshkevich V A, Vysloukh V A, Kartashev Y V. Optical surface waves at the interface between a linear dielectric and a photorefractive crystal[J]. Quantum Electronics, 30, 905-910(2000).

[38] Tian N, Lu K Q, Zhang X F et al. Influence of guiding parameters and propagation constants on photorefractive surface waves[J]. Chinese Journal of Lasers, 39, 0117002(2012).

[39] Chen W J, Lu K Q, Hui J L et al. Localized surface waves at the interface between linear dielectric and biased centrosymmetric photorefractive crystals[J]. Optics Express, 21, 15595-15602(2013).

[40] Tan C, Lei T, Zou M et al. Propagation dynamics of the circular Airy beam in the fractional Schrödinger equation under three kinds of potentials[J]. Optics Communications, 557, 130358(2024).

[41] Wang X Z, Tang F, Yuan M J et al. Experimental simulation of circular-Airy beam drift in atmospheric turbulence[J]. Chinese Journal of Lasers, 42, 0813001(2015).