| 171 | 0 | 70 |
| 下载次数 | 被引频次 | 阅读次数 |
孤子理论和实验研究涉及的非线性波模型大多数是整数阶导数，在分数阶导数非线性波模型中研究孤子存在性、稳定性及传输特性，进一步将孤子的研究拓展到分数维度。此外，分数衍射和分数色散效应为孤子传输的调控提供了新的自由度。本文简要介绍了分数量子力学的理论研究进展，概述了光学领域中分数薛定谔方程的理论和实验方面的研究进展，回顾了分数阶非线性薛定谔方程中孤子的数值算法、孤子解的存在形式、自发对称破缺以及演化，具体介绍了宇称时间对称的分数阶非线性薛定谔方程中孤子的自发对称破缺和鬼态的产生机制，并对该领域未来的发展做了展望。
Abstract：Most of the nonlinear wave models involved in soliton theory and experimental research are integer derivatives. The study of of solitons has been further extended from the existence, the stability and the transmission characteristics of solitons in the fractional derivative nonlinear wave model to fractional dimension. In addition, fractional diffraction and fractional dispersion effects provide a new freedom to control the soliton evolutions. This article briefly introduces the theoretical research progress of fractional quantum mechanics, outlines the theoretical and experimental research progress of the fractional Schr?dinger equation in the field of optics, and reviews the numerical algorithm of solitons in the fractional nonlinear Schr?dinger equation, the existence form of soliton solutions, and spontaneous symmetry breaking and evolutions. It also introduces in details the spontaneous symmetry breaking of solitons and the generation mechanism of ghost states in the fractional nonlinear Schr?dinger equation with parity-time symmetric potentials, and gives an outlook for the future development of this field.
[1] OLDHAM K B,SPANIER J. The Fractional Calculus[M]. New York:Academic Press, 1974.
[2] LASKIN N. Fractional quantum mechanics[J]. Physical Review E, 2000, 62(3):3135-3145. DOI:10.1103/PhysRevE.62.3135.
[3] LASKIN N. Fractional quantum mechanics and Lévy path integrals[J]. Physics Letters A, 2000, 268(4-6):398-305. DOI:10.1016/S0375-9601(00)00201-2.
[4] LASKIN N. Fractional Schr?dinger equation[J]. Physical Review E, 2002, 66(5):056108. DOI:10.1103/PhysRevE.66.056108.
[5] PODLUBNY I. Fractional Differential Equations[M].San Diego:Academic Press, 1999.
[6] KILBASA A, Srivastava H M, Trujillo J J. Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier, 2006.
[7] TARASOV V E. Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles, Fields and Media[M]. Berlin:Springer, 2010.
[8] HERRMANN R. Fractional Calculus:An Introduction for Physicists[M]. Singapore:World Scientific Publishing Co. Pte. Ltd., 2014.
[9] HILFERR. Applications of Fractional Calculus in Physics[M]. Singapore:World Scientific Publishing Co. Pte.Ltd., 2000.
[10] WEI Y C. Comment on “Fractional quantum mechanics” and “Fractional Schr?dinger equation”[J]. Physical Review E, 2016, 93(6):066103. DOI:10.1103/PhysRevE.93.066103.
[11] LASKINN. Reply to “Comment on ‘Fractional quantum mechanics’ and ‘Fractional Schr?dinger equation’”[J]. Physical Review E, 2016, 93(6):066104. DOI:10.1103/PhysRevE.93.066104.
[12] DONGJ P,XU M Y. Some solutions to the space fractional Schr?dinger equation using momentum representation method[J]. Journal of Mathematical Physics,2007, 48(7):072105.DOI:10.1063/1.2749172
[13] OLIVEIRA E C D, VAZ J. Tunneling in fractional quantum mechanics[J]. Journal of Physics A-Mathematical and Theoretical, 2011, 44(18):185303. DOI:10.1088/1751-8113/44/18/185303.
[14] LUCHKO Y. Fractional Schr?dinger equation for a particle moving in a potential well[J]. Journal of Mathematical Physics, 2013, 54(1):012111. DOI:10.1063/1.4777472.
[15] ZABA M, GARBACZEWSKI P. Solving fractional Schr?dinger-type spectral problems:Cauchy oscillator and Cauchy well[J]. Journal of Mathematical Physics, 2014, 55(9):092103.DOI:10.1063/1.4894057.
[16] TARE J D, ESGUERRA J P H. Transmission through locally periodic potentials in space-fractional quantum mechanics[J]. Physica A-Statistical Mechanics and Its Applications, 2014, 407:43-53. DOI:10.1016/j.physa.2014.03.084.
[17] ZHANG Y Q, LIU X, BELI?M R,et al. Propagation dynamics of a light beam in a fractional Schr?dinger equation[J]. Physical Review Letters, 2015, 115(18):180403.DOI:10.1103/PhysRevLett.115.180403.
[18] ZHANG Y Q, ZHONG H, BELI?M R,et al.Diffraction-free beams in fractional Schr?dinger equation[J]. Scientific Reports, 2016, 6(1):23645.DOI:10.1038/srep23645.
[19] HUANG X W, DENG Z X, FU X Q. Diffractionfree beams in fractional Schr?dinger equation with a linear potential[J]. Journal of the Optical Society of America B, 2017, 34(5):976-982. DOI:10.1364/JOSAB.34.000976.
[20] HUANG X W, DENG Z X, SHI X H, et al. Propagation characteristics of ring Airy beams modeled by the fractional Schr?dinger equation[J]. Journal of the Optical Society of America B, 2017, 34(10):2190-2197. DOI:10.1364/JOSAB.34.002190.
[21] HUANG X W, DENG Z X, BAI Y F,et al. Potential barrier-induced dynamics of finite energy Airy beams in fractional Schr?dinger equation[J]. Optics Express,2017, 25(26):32560-32569. DOI:10.1364/OE. 25.032560.
[22] ZHANG D, ZHANG Y Q, ZHANG Z Y, et al. Unveiling the link between fractional Schr?dinger equation and light propagation in honeycomb lattice[J]. Annalen der Physik, 2017, 529:1700149.DOI:10.1002/andp.201700149.
[23] ZHANG L F, LI C X, ZHONG H Z, et al. Propagation dynamics of super-Gaussian beams in fractional Schr?dinger equation:from linear to nonlinear regimes[J]. Optics Express, 2016, 24(13):14406-14418.DOI:10.1364/OE.24.014406.
[24] ZANG F, WANG Y, LI L. Dynamics of Gaussian beam modeled by fractional Schr?dinger equation with a variable coefficient[J]. Optics Express, 2018, 26(18):23740-2375.DOI:10.1364/OE.26.023740.
[25] HUANG C M, DONG L W. Beam propagation management in a fractional Schr?dinger equation[J]. Scientific Reports, 2017, 7(1):5442. DOI:10.1038/s41598-017-05926-5.
[26] WU Z K, YANG K B, REN X J, et al. Conical diffraction modulation in fractional dimensions with a PTsymmetric potential[J]. Chaos Solitons&Fractals,2022, 164:112631. DOI:10.1016/j.chaos.2022.112631.
[27] ZHANG Y Q, WANG R, ZHONG H, et al. Optical Bloch oscillation and Zener tunneling in the fractional Schr?dinger equation[J]. Scientific Reports, 2017, 7(1):17872. DOI:10.1038/s41598-017-17995-7.
[28] ZHANG Y Q, WANGR, ZHONG H, et al. Resonant mode conversions and Rabi oscillations in a fractional Schr?dinger equation[J]. Optics Express, 2017,25(25):32401-32410. DOI:10.1364/OE.25.032401.
[29] ZHANG Y Q, ZHONG H, BELI?M R, et al. PTsymmetry in a fractional Schr?dinger equation[J]. Laser&Photonics Review, 2016, 10(3):526-531.DOI:10.1002/lpor.201600037.
[30] LIS R, BAO Y Y, LIU Y H, et al. Bright solitons in fractional coupler with spatially periodical modulated nonlinearity[J]. Chaos Solitons&Fractals, 2022,162:112484. DOI:10.1016/j.chaos.2022.112484.
[31] ZENG L W, ZHU Y L, MALOMED B A, et al.Quadratic fractional solitons[J]. Chaos Solitons&Fractals, 2022, 154:111586. DOI:10.1016/j.chaos.2021.111586.
[32] WANG J F, JIN Y, GONGX G, et al. Generation of random soliton-like beams in a nonlinear fractional Schr?dinger equation[J]. Optics Express, 2022, 30(5):8199-8211. DOI:10.1364/OE.448972.
[33] BAO Y Y, LI S R, LIU Y H, et al. Gap solitons and nonlinear Bloch waves in fractional quantum coupler with periodic potential[J]. Chaos Solitons&Fractals,2022, 156:111853.DOI:10.1016/j.chaos.2022.111853.
[34] WANGQ, ZHANG L L, MALOMED B A, et al.Transformation of multipole and vortex solitons in the nonlocal nonlinear fractional Schr?dinger equation by means of Lévy-index management[J]. Chaos Solitons&Fractals, 2022, 157:111995. DOI:10.1016/j.chaos.2022.111995.
[35] ZHONG M, YAN Z Y. Formation of multi-peak gap solitons and stable excitations for double-Lévy-index and mixed fractional NLS equations with optical lattice potentials[J]. Proceedings of the Royal Society A,2023, 479:20230222. DOI:10.1098/rspa.2023.0222.
[36] ZHONG M, CHEN Y, YAN Z Y, et al. Suppression of soliton collapses, modulational instability and roguewave excitation in two-Lévy-index fractional Kerr media[J]. Proceedings of the Royal Society A, 2023,480:20230765. DOI:10.1098/rspa.2023.0765.
[37] SAKAGUCHI H,MALOMED B A. One-and twodimensional solitons in spin-orbit coupled Bose-Einstein condensates with fractional kinetic energy[J]. Journal of Physics B-Atomic Molecular and Optical Physics,2022, 55:155301. DOI:10.1088/1361-6455/ac7685.
[38] STICKLER B A. Potential condensed-matter realization of space-fractional quantum mechanics:The onedimensional Lévy crystal[J]. Physical Review E, 2013,88(1):012120. DOI:10.1103/PhysRevE.88.012120.
[39] PINSKER F, BAOW, ZHANG Y,et al. Fractional quantum mechanics in polariton condensates with velocity dependent mass[J]. Physical Review B 2015,92:195310. DOI:10.1103/PhysRevB.92.195310.
[40] LONGHI S. Fractional Schr?dinger equation in optics[J]. Optics Letters, 2015, 40(6):1117-1120. DOI:10.1364/OL.40.001117
[41] LIU S L,ZHANG Y W, MALOMED B A, et al.Experimental realisations of the fractional Schr?dinger equation in the temporal domain[J]. Nature Communications, 2023, 14(1):222. DOI:10.1038/s41467-023-35892-8.
[42] YANG J K. Newton-conjugate-gradient methods for solitary wave computations[J]. Journal of Computational Physics, 2009, 228(18):7007-7024.DOI:10.1016/j.jcp.2009.06.012.
[43] YANG J K. Nonlinear Waves in Integrable and Nonintegrable Systems[M]. America:SIAM, 2010.
[44] AGRAWAL G P. Nonlinear Fiber Optics[M]. Amsterdam:Elsevier, 2019.
[45] LI P F, MALOMED B A, MIHALACHE D. Symmetry breaking of spatial Kerr solitons in fractional dimension[J]. Chaos Solitons&Fractals, 2020, 132:109602. DOI:10.1016/j.chaos.2020.109602.
[46] LI P F, LI R J, DAI C Q. Existence, symmetry breaking bifurcation and stability of two-dimensional optical solitons supported by fractional diffraction[J].Optics Express, 2021, 29:3193-3210. DOI:10.1364/OE.415028.
[47] LI P F, MALOMED B A, MIHALACHE D. Vortex solitons in fractional nonlinear Schr?dinger equation with the cubic-quintic nonlinearity[J]. Chaos Solitons&Fractals,2020, 137:109783. DOI:10.1016/j.chaos.2020.109783.
[48] LI P F, MALOMED B A, MIHALACHE D.Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schr?dinger equation with a PTsymmetric potential[J]. Optics Letters, 2021, 46:3267-3270. DOI:10.1364/OL.428254.
[49] RODRIGUES A, LI K, ACHILLEOS V,et al. PTsymmetric double well potentials revisited:bifurcations,stability and dynamics[J]. Romanian Reports in Physics, 2013, 65:5-26. DOI:10.48550/arXiv.1207.1066.
[50] CARTARIUS H, HAAG D, DAST D,et al. Nonlinear Schr?dinger equation for a PT-symmetric delta-function double well[J]. Journal of Physics A-Mathematical and Theoretical, 2012, 45(44), 444008. DOI:10.1088/1751-8113/45/44/444008.
[51] SUSANTO H, KUSDIANTARA R, LI N, et al.Snakes and ghosts in a parity-time-symmetric chain of dimers[J]. Physical Review E, 2018, 97:062204.DOI:10.1103/PhysRevE.97.062204.
[52] ZHONG M,WANG L, LI P F,et al. Spontaneous symmetry breaking and ghost states supported by the fractional PT-symmetric saturable nonlinear Schr?dinger equation[J]. Chaos, 2023, 33(1):013106. DOI:10.1063/5.0128910.
[53] ZHONG M, YAN Z Y. Spontaneous symmetry breaking and ghost states in two-dimensional fractional nonlinear media with non-Hermitian potential[J]. Communications Physics, 2023, 6(1):92. DOI:10.1038/s42005-023-01212-1.
基本信息:
中图分类号:O437;O175.29
引用信息:
[1]李鹏飞,贺雪晴,蔡强.分数阶非线性波模型中孤子研究进展[J].新兴科学和技术趋势,2024,3(02):167-179.
2024-04-28
2024
2024-04-28
2024-04-28
2024
0
2024-07-11
2024-07-11
2024-07-11