Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices

doi:10.1007/s11467-023-1370-7

Front. Phys.

2024, Vol. 19

Issue (4) : 42201 https://doi.org/10.1007/s11467-023-1370-7

Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices

Xiuye Liu¹, Jianhua Zeng^1,^2,³(

)

¹. Center for Attosecond Science and Technology, State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences, Xi’an 710119, China
². University of Chinese Academy of Sciences, Beijing 100049, China
³. Collaborative lnnovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

Download: PDF(5206 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities. Moiré photonic and optical lattices — two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones — are a new emerging investigative tool for studying nonlinear localized waves of diverse types. Herein, a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction (linear nonlocality) and moiré optical lattices is investigated. Specifically, the flat-band feature is well preserved in shallow moiré optical lattices which, interact with the defocusing nonlinearity of the media, can support fundamental gap solitons, bound states composed of several fundamental solitons, and topological states (gap vortices) with vortex charge s = 1 and 2, all populated inside the finite gaps of the linear Bloch-wave spectrum. Employing the linear-stability analysis and direct perturbed simulations, the stability and instability properties of all the localized gap modes are surveyed, highlighting a wide stability region within the first gap and a limited one (to the central part) for the third gap. The findings enable insightful studies of highly localized gap modes in linear nonlocality (fractional) physical systems with shallow moiré patterns that exhibit extremely flat bands.

Keywords moiré optical lattices gap solitons and vortices ultracold atoms Gross−Pitaevskii/nonlinear fractional Schrödinger equation nonlinear fractional systems

Corresponding Author(s): Jianhua Zeng

Issue Date: 08 January 2024

Cite this article:

Xiuye Liu,Jianhua Zeng. Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices[J]. Front. Phys. , 2024, 19(4): 42201.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1370-7
https://academic.hep.com.cn/fop/EN/Y2024/V19/I4/42201

Fig.1 Band-gap structures of 2D moiré square optical lattices at different twisted angles

θ

in fractional systems. Contour shapes of the moiré lattice (shaded blue, lattice potential minima; shaded red, lattice potential maxima) at

θ = arctan ? (3 / 4)

(a) and

θ = arctan ? (5 / 12)

(b) in real space, the corresponding band-gap structures, expressed as chemical potential

μ

vs. Bloch quasi-momentum

K

in reciprocal lattice space, at

α = 1.6

are depicted in (c) and (d) respectively. (e) The first reduced Brillouin zone in the reciprocal lattice space; X, M, and

Γ

being the high-symmetry points. (f) Dependencies of

μ

on Lévy index

α

, and (g) on strength contrast

p

α = 1.3

, and

θ = arctan ? (3 / 4)

. Regions I, II and III represent respectively the first, second, and third band gaps.

Fig.2 Shapes of 2D fundamental gap solitons and the dependencies of norm

N

on chemical potential

μ

and Lévy index

α

in 2D moiré square optical lattices. (a?c) Characteristic shapes of fundamental gap solitons (GPs) at: (a)

μ = 2.5

N = 17.4

and

α = 1.3

; (b)

μ = 4.02

N = 144.1

and

α = 1.3

; (c)

μ = 2.6

N = 38.98

and

α = 0.4

. (d) Norm

N

vs. chemical potential

μ

, and (e) vs. Lévy index (

α

). The linear-stability eigenvalues of gap solitons, expressed by the largest real part of the perturbation growth rate,

λ (R)

, are shown as red dashed line in (d).

θ = arctan ? (3 / 4)

to all.

Fig.3 Shapes and norm

N

vs. chemical potential

μ

of bound-state GSs in 2D moiré square optical lattices in nonlinear fractional system with Lévy index

α = 1.3

. The 3D shapes (top), aerial views (central) and

N

vs.

μ

(bottom) of bound-state GSs. Parameters: (a)

μ = 2.2

N = 20.85

; (b)

μ = 3.47

N = 145.3

; (c)

μ = 2.2

N = 14.72

. Marked points in (d) correspond to bound states shown in (a), (b), and (c); the first two are quadruple gap modes, the latter is triple mode of structure.

Fig.4 Shapes and topological phases of vortex gap solitons with imprinted vorticity

s

in 2D moiré square optical lattices in nonlinear fractional system with Lévy index

α = 1.3

. The 3D shapes (top), aerial views (second line), phase structures (third line). Parameters: (a)

μ = 2.3

N = 31.67

; (b)

μ = 3.5

N = 306.52

; (c)

μ = 2.4

N = 133.78

; (d)

μ = 3.85

N = 1306

. Vorticity

s = 1

for (a, b) and

s = 2

for (c, d). The vortex solutions in (a, b, c, d) correspond respectively to the marked points (G, H, I, J) from Fig.5(a).

Fig.5 Norm

N

vs. chemical potential

μ

(a) (

α = 1.3

) and Lévy index

α

(b) (

μ = 2.6

) of vortex gap solitons with imprinted vorticity

s

. Other parameters: (a) vorticity (topological charge)

s = 1

for black line and

s = 2

for red dashed line; (b)

s = 1

Fig.6 Dynamics of fundamental GSs (a, b), higher-order GSs (c, d), and gap vortices with vorticity

s = 1

(e, f) supported by the 2D moiré square optical lattices. Other parameters: (a)

μ = 2.5, N = 13.61

; (b)

μ = 4.02, N = 140.9

; (c)

μ = 2.2, N = 20.85

; (d)

μ = 3.47, N = 145.3

; (e)

μ = 2.3, N = 31.67

; (f)

μ = 3.5, N = 306.52

Fig.7 Contour shapes of the moiré lattice (shaded blue, lattice potential minima; shaded yellow, lattice potential maxima) at

θ = arctan ? (3 / 4) ? 0.1

(a),

θ = arctan ? (3 / 4)

(b) and

θ = arctan ? (3 / 4) + 0.1

(c). Perturbed evolution in real time of stable (d) and unstable (e) fundamental gap solitons supported by non-periodic moiré optical lattices under the non-Pythagorean angle

θ = arctan ? (3 / 4) ? 0.1

. Other parameters: (d)

μ = 2.4, N = 5.59

; (e)

μ = 3.05, N = 39.34

α = 1.3

for all.

1	S. Kivshar Y.P. Agrawal G., Optical Solitons: From Fibers to Photonic Crystals, Academic, San Diego, 2003
2	Morsch O. , Oberthaler M. . Dynamics of Bose‒Einstein condensates in optical lattices. Rev. Mod. Phys., 2006, 78(1): 179 https://doi.org/10.1103/RevModPhys.78.179
3	V. Kartashov Y. , A. Malomed B. , Torner L. . Solitons in nonlinear lattices. Rev. Mod. Phys., 2011, 83(1): 247 https://doi.org/10.1103/RevModPhys.83.247
4	L. Garanovich I. , Longhi S. , A. Sukhorukov A. , S. Kivshar Y. . Light propagation and localization in modulated photonic lattices and waveguides. Phys. Rep., 2012, 518(1‒2): 1 https://doi.org/10.1016/j.physrep.2012.03.005
5	V. Konotop V. , Yang J. , A. Zezyulin D. . Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys., 2016, 88(3): 035002 https://doi.org/10.1103/RevModPhys.88.035002
6	V. Kartashov Y. , E. Astrakharchik G. , A. Malomed B. , Torner L. . Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat. Rev. Phys., 2019, 1(3): 185 https://doi.org/10.1038/s42254-019-0025-7
7	J. Eggleton B. , E. Slusher R. , M. de Sterke C. , A. Krug P. , E. Sipe J. . Bragg grating solitons. Phys. Rev. Lett., 1996, 76(10): 1627 https://doi.org/10.1103/PhysRevLett.76.1627
8	Mandelik D. , Morandotti R. , S. Aitchison J. , Silberberg Y. . Gap solitons in waveguide arrays. Phys. Rev. Lett., 2004, 92(9): 093904 https://doi.org/10.1103/PhysRevLett.92.093904
9	Peleg O. , Bartal G. , Freedman B. , Manela O. , Segev M. , N. Christodoulides D. . Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett., 2007, 98(10): 103901 https://doi.org/10.1103/PhysRevLett.98.103901
10	Eiermann B. , Anker T. , Albiez M. , Taglieber M. , Treutlein P. , P. Marzlin K. , K. Oberthaler M. . Bright Bose‒Einstein gap solitons of atoms with repulsive interaction. Phys. Rev. Lett., 2004, 92(23): 230401 https://doi.org/10.1103/PhysRevLett.92.230401
11	Anker Th. , Albiez M. , Gati R. , Hunsmann S. , Eiermann B. , Trombettoni A. , K. Oberthaler M. . Nonlinear self-trapping of matter waves in periodic potentials. Phys. Rev. Lett., 2005, 94(2): 020403 https://doi.org/10.1103/PhysRevLett.94.020403
12	H. Bennet F. , J. Alexander T. , Haslinger F. , Mitchell A. , N. Neshev D. , S. Kivshar Y. . Observation of nonlinear self-trapping of broad beams in defocusing waveguide arrays. Phys. Rev. Lett., 2011, 106(9): 093901 https://doi.org/10.1103/PhysRevLett.106.093901
13	Bersch C. , Onishchukov G. , Peschel U. . Optical gap solitons and truncated nonlinear Bloch waves in temporal lattices. Phys. Rev. Lett., 2012, 109(9): 093903 https://doi.org/10.1103/PhysRevLett.109.093903
14	Zeng L. , Zeng J. . Gap-type dark localized modes in a Bose‒Einstein condensate with optical lattices. Adv. Photonics, 2019, 1(4): 046004 https://doi.org/10.1117/1.AP.1.4.046004
15	Shi J. , Zeng J. . Self-trapped spatially localized states in combined linear-nonlinear periodic potentials. Front. Phys., 2020, 15(1): 12602 https://doi.org/10.1007/s11467-019-0930-3
16	Li J. , Zeng J. . Dark matter-wave gap solitons in dense ultra-cold atoms trapped by a one-dimensional optical lattice. Phys. Rev. A, 2021, 103(1): 013320 https://doi.org/10.1103/PhysRevA.103.013320
17	Chen J. , Zeng J. . Dark matter-wave gap solitons of Bose‒Einstein condensates trapped in optical lattices with competing cubic‒quintic nonlinearities. Chaos Solitons Fractals, 2021, 150: 111149 https://doi.org/10.1016/j.chaos.2021.111149
18	Chen Z. , Zeng J. . Localized gap modes of coherently trapped atoms in an optical lattice. Opt. Express, 2021, 29(3): 3011 https://doi.org/10.1364/OE.412554
19	Chen Z. , Zeng J. . Two-dimensional optical gap solitons and vortices in a coherent atomic ensemble loaded on optical lattices. Commun. Nonlinear Sci. Numer. Simul., 2021, 102: 105911 https://doi.org/10.1016/j.cnsns.2021.105911
20	Chen Z. , Zeng J. . Nonlinear localized modes in one-dimensional nanoscale dark-state optical lattices. Nanophotonics, 2022, 11(15): 3465 https://doi.org/10.1515/nanoph-2022-0213
21	Li J. , Zhang Y. , Zeng J. . Matter-wave gap solitons and vortices in three-dimensional parity‒time-symmetric optical lattices. iScience, 2022, 25(4): 104026 https://doi.org/10.1016/j.isci.2022.104026
22	Li J. , Zhang Y. , Zeng J. . 3D nonlinear localized gap modes in Bose‒Einstein condensates trapped by optical lattices and space-periodic nonlinear potentials. Adv. Photon. Res., 2022, 3(7): 2100288 https://doi.org/10.1002/adpr.202100288
23	Qin J. , Zhou L. . Supersolid gap soliton in a Bose‒Einstein condensate and optical ring cavity coupling system. Phys. Rev. E, 2022, 105(5): 054214 https://doi.org/10.1103/PhysRevE.105.054214
24	Yang J. , Zhang Y. . Spin‒orbit-coupled spinor gap solitons in Bose‒Einstein condensates. Phys. Rev. A, 2023, 107(2): 023316 https://doi.org/10.1103/PhysRevA.107.023316
25	Chen Z. , Wu Z. , Zeng J. . Light gap bullets in defocusing media with optical lattices. Chaos Solitons Fractals, 2023, 174: 113785 https://doi.org/10.1016/j.chaos.2023.113785
26	Huang C. , Dong L. . Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice. Opt. Lett., 2016, 41(24): 5636 https://doi.org/10.1364/OL.41.005636
27	Huang C. , Li C. , Deng H. , Dong L. . Gap Solitons in fractional dimensions with a quasi-periodic lattice. Ann. Phys., 2019, 531(9): 1900056 https://doi.org/10.1002/andp.201900056
28	Xie J. , Zhu X. , He Y. . Vector solitons in nonlinear fractional Schrödinger equations with parity‒time-symmetric optical lattices. Nonlinear Dyn., 2019, 97(2): 1287 https://doi.org/10.1007/s11071-019-05048-9
29	Zeng L. , Zeng J. . One-dimensional gap solitons in quintic and cubic‒quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential. Nonlinear Dyn., 2019, 98: 985 https://doi.org/10.1007/s11071-019-05240-x
30	Zeng L. , Zeng J. . Preventing critical collapse of higher-order solitons by tailoring unconventional optical diffraction and non-linearities. Commun. Phys., 2020, 3(1): 26 https://doi.org/10.1038/s42005-020-0291-9
31	Zhu X. , Yang F. , Cao S. , Xie J. , He Y. . Multipole gap solitons in fractional Schrödinger equation with parity‒time-symmetric optical lattices. Opt. Express, 2020, 28(2): 1631 https://doi.org/10.1364/OE.382876
32	Zeng L. , R. Belić M. , Mihalache D. , Shi J. , Li J. , Li S. , Lu X. , Cai Y. , Li J. . Families of gap solitons and their complexes in media with saturable nonlinearity and fractional diffraction. Nonlinear Dyn., 2022, 108(2): 1671 https://doi.org/10.1007/s11071-022-07291-z
33	Y. Bao Y. , R. Li S. , H. Liu Y. , F. Xu T. . Gap solitons and non-linear Bloch waves in fractional quantum coupler with periodic potential. Chaos Solitons Fractals, 2022, 156: 111853 https://doi.org/10.1016/j.chaos.2022.111853
34	Liu X. , A. Malomed B. , Zeng J. . Localized modes in nonlinear fractional systems with deep lattices. Adv. Theory Simul., 2022, 5(4): 2100482 https://doi.org/10.1002/adts.202100482
35	Dong L. , Huang C. . Double-hump solitons in fractional dimensions with a PT-symmetric potential. Opt. Express, 2018, 26(8): 10509 https://doi.org/10.1364/OE.26.010509
36	Huang C. , Dong L. . Beam propagation management in a fractional Schrödinger equation. Sci. Rep., 2017, 7(1): 5442 https://doi.org/10.1038/s41598-017-05926-5
37	Laskin N., Fractional Quantum Mechanics, World Scientific, 2018
38	A. Malomed B. . Optical solitons and vortices in fractional media: A mini-review of recent results. Photonics, 2021, 8(9): 353 https://doi.org/10.3390/photonics8090353
39	Liu S. , Zhang Y. , A. Malomed B. , Karimi E. . Experimental realizations of the fractional Schrödinger equation in the temporal domain. Nat. Commun., 2023, 14(1): 222 https://doi.org/10.1038/s41467-023-35892-8
40	Cao Y. , Fatemi V. , Fang S. , Watanabe K. , Taniguchi T. , Kaxiras E. , Jarillo-Herrero P. . Unconventional superconductivity in magic-angle graphene superlattices. Nature, 2018, 556(7699): 43 https://doi.org/10.1038/nature26160
41	Cao Y. , Fatemi V. , Demir A. , Fang S. , L. Tomarken S. , Y. Luo J. , D. Sanchez-Yamagishi J. , Watanabe K. , Taniguchi T. , Kaxiras E. , C. Ashoori R. , Jarillo-Herrero P. . Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature, 2018, 556(7699): 80 https://doi.org/10.1038/nature26154
42	Carr S. , Massatt D. , Fang S. , Cazeaux P. , Luskin M. , Kaxiras E. . Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle. Phys. Rev. B, 2017, 95(7): 075420 https://doi.org/10.1103/PhysRevB.95.075420
43	Huang C. , Ye F. , Chen X. , V. Kartashov Y. , V. Konotop V. , Torner L. . Localization‒delocalization wavepacket transition in Pythagorean aperiodic potentials. Sci. Rep., 2016, 6(1): 32546 https://doi.org/10.1038/srep32546
44	Wang P. , Zheng Y. , Chen X. , Huang C. , V. Kartashov Y. , Torner L. , V. Konotop V. , Ye F. . Localization and delocalization of light in photonic moiré lattices. Nature, 2020, 577(7788): 42 https://doi.org/10.1038/s41586-019-1851-6
45	Fu Q. , Wang P. , Huang C. , V. Kartashov Y. , Torner L. , V. Konotop V. , Ye F. . Optical soliton formation controlled by angle twisting in photonic moiré lattices. Nat. Photonics, 2020, 14(11): 663 https://doi.org/10.1038/s41566-020-0679-9
46	R. Mao X. , K. Shao Z. , Y. Luan H. , L. Wang S. , M. Ma R. . Magic-angle lasers in nanostructured moiré superlattice. Nat. Nanotechnol., 2021, 16(10): 1099 https://doi.org/10.1038/s41565-021-00956-7
47	V. Kartashov Y. , Ye F. , V. Konotop V. , Torner L. . Multi-frequency solitons in commensurate-incommensurate photonic moiré lattices. Phys. Rev. Lett., 2021, 127(16): 163902 https://doi.org/10.1103/PhysRevLett.127.163902
48	V. Kartashov Y. . Light bullets in moiré lattices. Opt. Lett., 2022, 47(17): 4528 https://doi.org/10.1364/OL.471022
49	K. Ivanov S. , V. Konotop V. , V. Kartashov Y. , Torner L. . Vortex solitons in moiré optical lattices. Opt. Lett., 2023, 48(14): 3797 https://doi.org/10.1364/OL.494681
50	A. Arkhipova A. , V. Kartashov Y. , K. Ivanov S. , A. Zhuravitskii S. , N. Skryabin N. , V. Dyakonov I. , A. Kalinkin A. , P. Kulik S. , O. Kompanets V. , V. Chekalin S. , Ye F. , V. Konotop V. , Torner L. , N. Zadkov V. . Observation of linear and nonlinear light localization at the edges of moiré arrays. Phys. Rev. Lett., 2023, 130(8): 083801 https://doi.org/10.1103/PhysRevLett.130.083801
51	S. Sunku S. , X. Ni G. , Y. Jiang B. , Yoo H. , Sternbach A. , S. McLeod A. , Stauber T. , Xiong L. , Taniguchi T. , Watanabe K. , Kim P. , M. Fogler M. , N. Basov D. . Photonic crystals for nano-light in moiré graphene superlattices. Science, 2018, 362(6419): 1153 https://doi.org/10.1126/science.aau5144
52	J. M. Kort-Kamp W. , J. Culchac F. , B. Capaz R. , A. Pinheiro F. . Photonic spin Hall effect in bilayer graphene moiré superlattices. Phys. Rev. B, 2018, 98(19): 195431 https://doi.org/10.1103/PhysRevB.98.195431
53	Hu G. , Ou Q. , Si G. , Wu Y. , Wu J. , Dai Z. , Krasnok A. , Mazor Y. , Zhang Q. , Bao Q. , W. Qiu C. , Alù A. . Topological polaritons and photonic magic angles in twisted α-MoO3 bilayers. Nature, 2020, 582(7811): 209 https://doi.org/10.1038/s41586-020-2359-9
54	Chen M. , Lin X. , H. Dinh T. , Zheng Z. , Shen J. , Ma Q. , Chen H. , Jarillo-Herrero P. , Dai S. . Configurable phonon polaritons in twisted α-MoO3. Nat. Mater., 2020, 19(12): 1307 https://doi.org/10.1038/s41563-020-0732-6
55	González-Tudela A. , I. Cirac J. . Cold atoms in twisted-bilayer optical potentials. Phys. Rev. A, 2019, 100(5): 053604 https://doi.org/10.1103/PhysRevA.100.053604
56	Salamon T. , Celi A. , W. Chhajlany R. , Frérot I. , Lewenstein M. , Tarruell L. , Rakshit D. . Simulating twistronics without a twist. Phys. Rev. Lett., 2020, 125(3): 030504 https://doi.org/10.1103/PhysRevLett.125.030504
57	W. Luo X. , Zhang C. . Spin-twisted optical lattices: Tunable flat bands and Larkin‒Ovchinnikov superfluids. Phys. Rev. Lett., 2021, 126(10): 103201 https://doi.org/10.1103/PhysRevLett.126.103201
58	Ning T. , Ren Y. , Huo Y. , Cai Y. . Efficient high harmonic generation in nonlinear photonic moiré superlattice. Front. Phys., 2023, 18(5): 52305 https://doi.org/10.1007/s11467-023-1296-0
59	Ma Z. , J. Chen W. , Chen Y. , H. Gao J. , C. Xie X. . Flat band localization due to self-localized orbital. Front. Phys., 2023, 18(6): 63302 https://doi.org/10.1007/s11467-023-1306-2
60	Chen Z. , Liu X. , Zeng J. . Electromagnetically induced moiré optical lattices in a coherent atomic gas. Front. Phys., 2022, 17(4): 42508 https://doi.org/10.1007/s11467-022-1153-6
61	Meng Z. , Wang L. , Han W. , Liu F. , Wen K. , Gao C. , Wang P. , Chin C. , Zhang J. . Atomic Bose‒Einstein condensate in twisted-bilayer optical lattices. Nature, 2023, 615(7951): 231 https://doi.org/10.1038/s41586-023-05695-4
62	Huang C. , Dong L. , Deng H. , Zhang X. , Gao P. . Fundamental and vortex gap solitons in quasiperiodic photonic lattices. Opt. Lett., 2021, 46(22): 5691 https://doi.org/10.1364/OL.443051
63	Liu X. , Zeng J. . Matter-wave gap solitons and vortices of dense Bose‒Einstein condensates in moiré optical lattices. Chaos Solitons Fractals, 2023, 174: 113869 https://doi.org/10.1016/j.chaos.2023.113869
64	Liu X. , Zeng J. . Gap solitons in parity‒time symmetric moiré optical lattices. Photon. Res., 2023, 11(2): 196 https://doi.org/10.1364/PRJ.474527
65	Yang J., Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM: Philadelphia, 2010
66	Cai M. , P. Li C. . On Riesz derivative. Fract. Calc. Appl. Anal., 2019, 22(2): 287 https://doi.org/10.1515/fca-2019-0019
67	Duo S. , Zhang Y. . Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications. Comput. Methods Appl. Mech. Eng., 2019, 355: 639 https://doi.org/10.1016/j.cma.2019.06.016
68	Laskin N. . Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A, 2000, 268(4−6): 298 https://doi.org/10.1016/S0375-9601(00)00201-2
69	Laskin N. . Fractional quantum mechanics. Phys. Rev. E, 2000, (3): 3135 https://doi.org/10.1103/PhysRevE.62.3135
70	Laskin N. . Fractional Schrödinger equation. Phys. Rev. E, 2002, 66(5): 056108 https://doi.org/10.1103/PhysRevE.66.056108
71	Zhang L. , Li C. , Zhong H. , Xu C. , Lei D. , Li Y. , Fan D. . Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: From linear to nonlinear regimes. Opt. Express, 2016, 24(13): 14406 https://doi.org/10.1364/OE.24.014406
72	Zhang L. , He Z. , Conti C. , Wang Z. , Hu Y. , Lei D. , Li Y. , Fan D. . Modulational instability in fractional nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul., 2017, 48: 531 https://doi.org/10.1016/j.cnsns.2017.01.019
73	Vakhitov M. , Kolokolov A. . Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron., 1973, 16(7): 783 https://doi.org/10.1007/BF01031343
74	Ferrando A. , Zacarés M. , A. García-March M. . Vorticity cutoff in nonlinear photonic crystals. Phys. Rev. Lett., 2005, 95(4): 043901 https://doi.org/10.1103/PhysRevLett.95.043901

[1]	Yanming Hu, Yifan Fei, Xiao-Long Chen, Yunbo Zhang. Collisional dynamics of symmetric two-dimensional quantum droplets[J]. Front. Phys. , 2022, 17(6): 61505-.
[2]	Zhiming Chen, Xiuye Liu, Jianhua Zeng. Electromagnetically induced moiré optical lattices in a coherent atomic gas[J]. Front. Phys. , 2022, 17(4): 42508-.
[3]	Dan-wei Zhang (张丹伟), Zi-dan Wang (汪子丹), Shi-liang Zhu (朱诗亮). Relativistic quantum effects of Dirac particles simulated by ultracold atoms[J]. Front. Phys. , 2012, 7(1): 31-53.
[4]	LIU Xiong-jun, LIU Xin, KWEK Leong-Chuan, OH ChooHiap. Manipulating atomic states via optical orbital angular-momentum[J]. Front. Phys. , 2008, 3(2): 113-125.

Viewed

Full text

Abstract

Cited

Shared

Discussed