Abstract

We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrödinger equation with the fractional Laplacian. Depending on the beam separation ($d$), relative phase and Lévy index ($\alpha $), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as $\alpha $ increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Lévy index exceeds a critical value ${\alpha _c}$, the period of breather soliton for $d > 0$ is orders of magnitude larger than for $d < 0$, while the opposite occurs as $\alpha < {\alpha _c}$. Our results reveal a novel scenario for Airy beams interaction in the framework of fractional nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (4)

2018 (3)

2017 (4)

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

M. Zhang, G. Huo, H. Zhong, and Z. Hui, “Interactions between self-accelerating beams in photorefractive media,” Opt. Express 25(18), 22104–22112 (2017).
[Crossref]

2016 (8)

M. Shen, W. Li, and R. K. Lee, “Control on the anomalous interactions of Airy beams in nematic liquid crystals,” Opt. Express 24(8), 8501–8511 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24(13), 14406–14418 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41(24), 5636–5639 (2016).
[Crossref]

2015 (4)

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
[Crossref]

S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40(6), 1117–1120 (2015).
[Crossref]

X. Zhong, X. Du, and K. Cheng, “Evolution of finite energy Airy pulses and soliton generation in optical fibers with cubic-quintic nonlinearity,” Opt. Express 23(23), 29467–29475 (2015).
[Crossref]

2014 (1)

2013 (2)

Y. Zhang, M. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4587 (2013).
[Crossref]

B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88(1), 012120 (2013).
[Crossref]

2011 (1)

2008 (1)

2007 (4)

2002 (1)

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66(5), 056108 (2002).
[Crossref]

2000 (2)

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62(3), 3135–3145 (2000).
[Crossref]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268(4-6), 298–305 (2000).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Ahmed, N.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Belic, M.

Belic, M. R.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22(6), 7160–7171 (2014).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Chen, H.

Chen, M.

M. Chen, Q. Guo, D. Lu, and W. Hu, “Variational approach for breathers in a nonlinear fractional Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simulat. 71(15), 73–81 (2019).
[Crossref]

Chen, Z.

Cheng, K.

Christodoulides, D. N.

N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

Conti, C.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dong, L.

Du, X.

Efremidis, N. K.

Fan, D.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24(13), 14406–14418 (2016).
[Crossref]

Fattal, Y.

Gao, J.

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
[Crossref]

Ge, L.

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
[Crossref]

Guo, Q.

M. Chen, Q. Guo, D. Lu, and W. Hu, “Variational approach for breathers in a nonlinear fractional Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simulat. 71(15), 73–81 (2019).
[Crossref]

Gutiérrez-Vega, J. C.

He, Z.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Herrmann, R.

R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2011).

Hu, W.

M. Chen, Q. Guo, D. Lu, and W. Hu, “Variational approach for breathers in a nonlinear fractional Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simulat. 71(15), 73–81 (2019).
[Crossref]

Hu, Y.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Huang, C.

Huang, T.

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

Hui, Z.

Huo, G.

Kienle, A.

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Laskin, N.

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66(5), 056108 (2002).
[Crossref]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62(3), 3135–3145 (2000).
[Crossref]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268(4-6), 298–305 (2000).
[Crossref]

Lee, R. K.

Lei, D.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24(13), 14406–14418 (2016).
[Crossref]

Li, C.

Li, F.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Li, J.

Li, L.

Li, W.

Li, Y.

Liemert, A.

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Liu, X.

X. Yao and X. Liu, “Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential,” Photonics Res. 6(9), 875–879 (2018).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Longhi, S.

Lu, D.

M. Chen, Q. Guo, D. Lu, and W. Hu, “Variational approach for breathers in a nonlinear fractional Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simulat. 71(15), 73–81 (2019).
[Crossref]

Lu, K.

Malomed, B. A.

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Marom, D. M.

Ortigueira, M. D.

M. D. Ortigueira, Fractional Calculus for Scientists and Engineers (Springer Science, 2011).

Podlubny, I.

I. Podlubny, Fractional Differential Equations (Academic, 1999).

Rudnick, A.

Segev, M.

Shang, C.

Shen, M.

M. Shen, W. Li, and R. K. Lee, “Control on the anomalous interactions of Airy beams in nematic liquid crystals,” Opt. Express 24(8), 8501–8511 (2016).
[Crossref]

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
[Crossref]

Siviloglou, G. A.

Stickler, B. A.

B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88(1), 012120 (2013).
[Crossref]

Wang, R.

Wang, Y.

Wang, Z.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Wu, Z.

Xiao, M.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Xu, C.

Yao, X.

X. Yao and X. Liu, “Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential,” Photonics Res. 6(9), 875–879 (2018).
[Crossref]

Ye, F.

Zang, F.

Zeng, J.

Zeng, L.

Zhang, D.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Zhang, J.

Zhang, L.

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24(13), 14406–14418 (2016).
[Crossref]

Zhang, M.

Zhang, Y.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22(6), 7160–7171 (2014).
[Crossref]

Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22(6), 7160–7171 (2014).
[Crossref]

Y. Zhang, M. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4587 (2013).
[Crossref]

Y. Zhang, M. Belić, Z. Wu, H. Zheng, K. Lu, Y. Li, and Y. Zhang, “Soliton pair generation in the interactions of airy and nonlinear accelerating beams,” Opt. Lett. 38(22), 4585–4587 (2013).
[Crossref]

Zhang, Z.

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Zheng, H.

Zhong, H.

Zhong, W.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

Zhong, X.

Zhu, Y.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

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[Crossref]

Ann. Phys. (2)

W. Zhong, M. R. Belić, B. A. Malomed, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
[Crossref]

D. Zhang, Y. Zhang, Z. Zhang, N. Ahmed, Y. Zhang, F. Li, M. R. Belić, and M. Xiao, “Unveiling the link between fractional Schrödinger equation and light propagation in honeycomb lattice,” Ann. Phys. 529(9), 1700149 (2017).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (1)

L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simul. 48, 531–540 (2017).
[Crossref]

Commun. Nonlinear Sci. Numer. Simulat. (1)

M. Chen, Q. Guo, D. Lu, and W. Hu, “Variational approach for breathers in a nonlinear fractional Schrödinger equation,” Commun. Nonlinear Sci. Numer. Simulat. 71(15), 73–81 (2019).
[Crossref]

Laser Photonics Rev. (1)

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Mathematics (1)

A. Liemert and A. Kienle, “Fractional Schrödinger equation in the presence of the linear potential,” Mathematics 4(2), 31 (2016).
[Crossref]

Opt. Express (12)

L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a PT symmetric potential,” Opt. Express 26(8), 10509–10518 (2018).
[Crossref]

F. Zang, Y. Wang, and L. Li, “Dynamics of gaussian beam modeled by fractional schrödinger equation with a variable coefficient,” Opt. Express 26(18), 23740–23750 (2018).
[Crossref]

X. Zhong, X. Du, and K. Cheng, “Evolution of finite energy Airy pulses and soliton generation in optical fibers with cubic-quintic nonlinearity,” Opt. Express 23(23), 29467–29475 (2015).
[Crossref]

Y. Zhang, M. R. Belić, H. Zheng, H. Chen, C. Li, Y. Li, and Y. Zhang, “Interactions of Airy beams, nonlinear accelerating beams, and induced solitons in Kerr and saturable nonlinear media,” Opt. Express 22(6), 7160–7171 (2014).
[Crossref]

M. Zhang, G. Huo, H. Zhong, and Z. Hui, “Interactions between self-accelerating beams in photorefractive media,” Opt. Express 25(18), 22104–22112 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017).
[Crossref]

C. Huang, C. Shang, J. Li, L. Dong, and F. Ye, “Localization and Anderson delocalization of light in fractional dimensions with a quasi-periodic lattice,” Opt. Express 27(5), 6259–6267 (2019).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24(13), 14406–14418 (2016).
[Crossref]

J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre Gaussian Modes,” Opt. Express 15(10), 6300–6313 (2007).
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J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18), 17298–17307 (2011).
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M. Shen, W. Li, and R. K. Lee, “Control on the anomalous interactions of Airy beams in nematic liquid crystals,” Opt. Express 24(8), 8501–8511 (2016).
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Opt. Lett. (6)

Optica (1)

Photonics Res. (1)

X. Yao and X. Liu, “Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential,” Photonics Res. 6(9), 875–879 (2018).
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B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88(1), 012120 (2013).
[Crossref]

W. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94(1), 012216 (2016).
[Crossref]

Phys. Rev. Lett. (2)

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
[Crossref]

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Sci. Rep. (2)

M. Shen, J. Gao, and L. Ge, “Solitons shedding from Airy beams and bound states of breathing Airy solitons in nonlocal nonlinear media,” Sci. Rep. 5(1), 9814 (2015).
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Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6(1), 23645 (2016).
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Figures (7)

Fig. 1.
Fig. 1. Spatial evolution of the interaction of two in-phase Airy (top row) and sech beams (bottom row) with different initial interval d in the FNLSE.
Fig. 2.
Fig. 2. (a, c) The evolution of maximum values of ${|U |^{0.5}}$ versus propagation distance and (b, d) the incident beam profiles, which consist in the superposition of two in-phase beams with different initial intervals for (a, b) Airy and (c, d) sech beams.
Fig. 3.
Fig. 3. Beam shapes from the coherent superposition of two Airy (a, c) and sech (b, d) beams as a function of relative separation for the cases of (a, b) in phase and (c, d) out of phase. Black dashed lines denote the position of peak intensity.
Fig. 4.
Fig. 4. (a, b) Maximum amplitude and (c) on-axis amplitude of coherent superposition of two in-phase (a, c) and out-of-phase (b) beams as a function of their relative separation. (d) The period of breathing soliton formed from in-phase beams interaction.
Fig. 5.
Fig. 5. The interactions of two in-phase Airy beams with $A = 3$ for different initial interval and different Lévy index $\alpha $ in the FNLSE.
Fig. 6.
Fig. 6. (a) Period and (b) maximal values of ${|U |^{0.5}}$ of the breathing soliton for varying $\alpha $ Dashed green line in (a) denotes the critical point of the Lévy index. Inset in (a) a zoomed range.
Fig. 7.
Fig. 7. The interactions of two out-of-phase Airy beams with $A = 3$ for different initial interval d and different Lévy index $\alpha $ in the FNLSE.

Tables (1)

Tables Icon

Table 1. Peak intensity and its interval for different initial relative spacing

Equations (8)

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i U Z = ( 2 X 2 ) α / α 2 2 U N 2 | U | 2 U ,
i U ^ Z = | κ | α U ^ N 2 | U ^ | 2 U ^ ,
U ( Z , X ) = 1 2 π U ^ ( 0 , κ ) exp ( i | κ | α Z ) exp ( i κ Z ) d κ ,
U ( 0 , X ) = A [ ? o x y i n s e r t s t a r t a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090548 0400 " ? U ? o x y i n s e r t e n d ? ? o x y d e l e t e a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090548 0400 " c o n t e n t =" A " ? i ? o x y i n s e r t s t a r t a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090557 0400 " ? n ? o x y i n s e r t e n d ? ( X + d ) + ? o x y i n s e r t s t a r t a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090602 0400 " ? U ? o x y i n s e r t e n d ? ? o x y d e l e t e a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090601 0400 " c o n t e n t =" A " ? i ? o x y i n s e r t s t a r t a u t h o r =" k i v a n o v a " t i m e s t a m p =" 20190918 T 090606 0400 " ? n ? o x y i n s e r t e n d ? ( X + d ) exp ( i θ ) ] ,
U ( 0 , X ) = A [ A i ( X + d ) + A i ( X + d ) exp ( i θ ) ] ,
U a i r y ( X , d ) = { 2 A π 0 cos ( d t + t 3 3 ) cos ( X t ) d t ,   for   θ   =   0 2 A π 0 sin ( d t + t 3 3 ) sin ( X t ) d t ,   for   θ   =   π ,
U sech ( X , d ) = { 2 ( e X + e X ) ( e d + e d ) e 2 X + e 2 X + e 2 d + e 2 d , for  θ   =   0 2 ( e X e X ) ( e d e d ) e 2 X + e 2 X + e 2 d + e 2 d , for  θ   =   π ,
U X = 0 ( d ) = { 2 A π 0 cos ( d t + t 3 3 ) d t ,  for  A i r y   b e a m 4 e d + e d   ,  for sech  b e a m ,

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