Abstract

We investigate the properties of double-hump solitons supported by the nonlinear Schrödinger equation featuring a combination of parity-time symmetry and fractional-order diffraction effect. Two classes of nonlinear states, i.e., out-of-phase and in-phase solitons are found. Each class contains two families of solitons originating from the same linear mode in both focusing and defocusing nonlinear Kerr media. The critical phase-transition point increases monotonously with increasing Lévy index. For strong gain and loss, out-of-phase solitons in focusing media are stable in a wide parameter window and are almost completely unstable in media with a defocusing nonlinearity. The stability of in-phase solitons is opposite to that of out-of-phase solitons. In-phase solitons in defocusing media are stable in their entire existence domains provided that the gain-loss strength is below a critical value. Meanwhile, the stability region shrinks with the decrease of Lévy index. We, thus, put forward the first example of spatial solitons in fractional dimensions with a parity-time symmetry.

© 2018 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]
  33. W.-P. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
    [Crossref]
  34. W.-P. Zhong, M. R. Belić, B. A. Malomed, Y. Zhang, and T. Huang, “Spatiotemporal accessible solitons in fractional dimensions,” Phys. Rev. E 94, 012216 (2016).
    [Crossref] [PubMed]
  35. C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
    [Crossref]
  36. B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
    [Crossref]
  37. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).
    [Crossref] [PubMed]
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    [Crossref]

2017 (1)

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

2016 (9)

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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

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

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

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the Quantum Fractional Oscillator,” J. Phys.: Condens. Matter. 698(1), 012025 (2016).

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

S. Nixon and J. Yang, “Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides,” Opt. Lett. 41(12), 2747–2750 (2016).
[Crossref] [PubMed]

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in 𝒫𝒯-symmetric systems,” Rev. Mod. Phys. 88, 035002 (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] [PubMed]

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] [PubMed]

2015 (3)

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

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological States in Partially-𝒫𝒯-Symmetric Azimuthal Potentials,” Phys. Rev. Lett. 115, 193902 (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, 180403 (2015).
[Crossref]

2014 (6)

2013 (3)

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5(1), 83–130 (2013).
[Crossref]

A. Kundu and B. Seradjeh, “Transport Signatures of Floquet Majorana Fermions in Driven Topological Superconductors,” Phys. Rev. Lett. 111, 136402 (2013).
[Crossref] [PubMed]

2012 (3)

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
[Crossref]

2011 (2)

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[Crossref]

2010 (2)

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
[Crossref] [PubMed]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2002 (2)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

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

2000 (2)

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

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

1998 (1)

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

1983 (1)

R. B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[Crossref]

Ahmed, N.

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

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Alberucci, A.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in 𝒫𝒯-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
[Crossref]

Assanto, G.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in 𝒫𝒯-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
[Crossref]

Belic, M.

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

Belic, M. R.

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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

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

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (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, 180403 (2015).
[Crossref]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

Brazhnyi, V. A.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in 𝒫𝒯-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Chen, X.

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Christodoulides, D. N.

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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Dong, L.

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[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] [PubMed]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Fan, D.

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
[Crossref]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Guo, B.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

Guo, Z.

He, Y.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Huang, C.

Huang, D.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

Huang, T.

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

Jiang, X.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Jisha, C. P.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in 𝒫𝒯-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Konotop, V. V.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in 𝒫𝒯-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological States in Partially-𝒫𝒯-Symmetric Azimuthal Potentials,” Phys. Rev. Lett. 115, 193902 (2015).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[Crossref]

Kundu, A.

A. Kundu and B. Seradjeh, “Transport Signatures of Floquet Majorana Fermions in Driven Topological Superconductors,” Phys. Rev. Lett. 111, 136402 (2013).
[Crossref] [PubMed]

Laskin, N.

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

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

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

Laughlin, R. B.

R. B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

Lei, D.

Li, C.

Li, H.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Li, Y.

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Liu, S.

Liu, X.

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, 180403 (2015).
[Crossref]

Longhi, S.

Lumer, Y.

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Malomed, B. A.

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Mikael, C.

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

Nixon, S.

S. Nixon and J. Yang, “Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides,” Opt. Lett. 41(12), 2747–2750 (2016).
[Crossref] [PubMed]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
[Crossref]

Olivar-Romero, F.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the Quantum Fractional Oscillator,” J. Phys.: Condens. Matter. 698(1), 012025 (2016).

Plotnik, Y.

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Rechtsman, M.

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Rosas-Ortiz, O.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the Quantum Fractional Oscillator,” J. Phys.: Condens. Matter. 698(1), 012025 (2016).

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Segev, M.

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Seradjeh, B.

A. Kundu and B. Seradjeh, “Transport Signatures of Floquet Majorana Fermions in Driven Topological Superconductors,” Phys. Rev. Lett. 111, 136402 (2013).
[Crossref] [PubMed]

Shi, Z.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Torner, L.

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

Wang, J.

Wen, J.

Xiao, M.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 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 Photon. 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, 180403 (2015).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5(1), 83–130 (2013).
[Crossref]

Xu, C.

Yang, J.

Ye, F.

Zezyulin, D. A.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in 𝒫𝒯-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[Crossref]

Zhang, L.

Zhang, Y.

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

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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

W.-P. Zhong, M. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (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 Photon. 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, 23645 (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, 23645 (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, 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, 180403 (2015).
[Crossref]

J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5(1), 83–130 (2013).
[Crossref]

Zhong, H.

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 Photon. Rev. 10(3), 526–531 (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] [PubMed]

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

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 Photon. 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, 180403 (2015).
[Crossref]

Zhong, W.-P.

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

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

Zhou, K.

Zhu, X.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Adv. Opt. Photon. (1)

Ann. Phys. (1)

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

J. Math. Phys. (1)

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

J. Phys.: Condens. Matter. (1)

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the Quantum Fractional Oscillator,” J. Phys.: Condens. Matter. 698(1), 012025 (2016).

Laser Photon. 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 Photon. Rev. 10(3), 526–531 (2016).
[Crossref]

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (9)

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] [PubMed]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39(19), 5641–5644 (2014).
[Crossref] [PubMed]

C. Huang, F. Ye, Y. V. Kartashov, B. A. Malomed, and X. Chen, “PT symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
[Crossref]

J. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39(19), 5547–5550 (2014).
[Crossref] [PubMed]

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

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).
[Crossref] [PubMed]

S. Nixon and J. Yang, “Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides,” Opt. Lett. 41(12), 2747–2750 (2016).
[Crossref] [PubMed]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
[Crossref] [PubMed]

J. Yang, “Partially PT symmetric optical potentials with all-real spectra and soliton families in multidimensions,” Opt. Lett. 39(5), 1133–1136 (2014).
[Crossref] [PubMed]

Phys. Lett. A (1)

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

Phys. Rev. A (5)

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in 𝒫𝒯-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
[Crossref]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
[Crossref]

Phys. Rev. E (3)

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

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

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

Phys. Rev. Lett. (10)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex Extension of Quantum Mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-Symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref] [PubMed]

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, 180403 (2015).
[Crossref]

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological States in Partially-𝒫𝒯-Symmetric Azimuthal Potentials,” Phys. Rev. Lett. 115, 193902 (2015).
[Crossref]

R. B. Laughlin, “Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

A. Kundu and B. Seradjeh, “Transport Signatures of Floquet Majorana Fermions in Driven Topological Superconductors,” Phys. Rev. Lett. 111, 136402 (2013).
[Crossref] [PubMed]

Y. Lumer, Y. Plotnik, M. Rechtsman, C. Mikael, and M. Segev, “Nonlinearly Induced PT Transition in Photonic Systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Rev. Mod. Phys. (1)

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in 𝒫𝒯-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
[Crossref]

Sci. Rep. (2)

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

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

Other (1)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, Philadelphia, 2010).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a) Profile of potential. (b) Critical phase-transition point versus the Lévy index. Real (top) and imaginary (bottom) parts of the first two eigenvalues of linear eigenmodes in systems with α = 1.3 (c) and 1.7 (d).
Fig. 2
Fig. 2 Profiles of out-of-phase solitons in media with a defocusing (a, c) and a focusing (b, d) nonlinearity. (c) b = 2.4. (d) b = 3.6. α = 1.7 in (a–c) and χ = 1 in (a, b, d).
Fig. 3
Fig. 3 (a) Power of solitons in defocusing (left) and focusing (right) media originating from the same linear mode. Solid: stable; dotted: unstable. (b) Top: the ratio between the maxima of imaginary and real parts; Bottom: the effective width of solitons. (c, d) Instability growth rate versus the propagation constant. α = 1.7 in (a–c) and 1.3 in (d). χ = 1 in all the panels.
Fig. 4
Fig. 4 Profiles of in-phase solitons in media with a defocusing (a, c) and a focusing (b, d) nonlinearity. (c) b = 2.7. (d) b = 3.6. χ = 1 in (a, b, d) and α = 1.7 in (a, b, c).
Fig. 5
Fig. 5 (a) Dependence of power of solitons in defocusing (left) and focusing (right) media originating from the same linear mode on the propagation constant. Solid: stable; dotted: unstable. (b) The existence domain of solitons in defocusing Kerr media. (c) Instability growth rate versus the propagation constant for solitons in systems with different χ. (d) Spectrum of the linearization operator for solitons marked by the blue circles in (c) at b = 1.23 (top) and 2.48 (bottom). α = 1.7 in all the panels.
Fig. 6
Fig. 6 Evolutions of out-of-phase solitons (a, b) marked in Fig. 3(c) and (c) marked in Fig. 3(d). (d–f) Propagation of in-phase solitons marked in Fig. 5(c). (a) b = 2.8, z = 600. (b) b = 4.2, z = 6000. (c) b = 3.18, z = 6000. (d) b = 1.23, z = 60. (e) b = 2.48, z = 6000. (f) b = 1.71, z = 6000. χ = 1 in (a–e) and 0.6 in (f). (b, c) corresponds to the focusing case and others corresponds to the defocusing case. α = 1.3 in (c) and 1.7 in other panels. x belongs to [−7, 7] and |w| is shown in (a–f).

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

i q z = 1 2 ( 2 x 2 ) α / 2 q + γ | q | 2 q V ( x ) q ,
1 2 ( d 2 d x 2 ) α / 2 w + b w [ V r ( x ) + i χ V i ( x ) ] w + γ | w | 2 w = 0 ,
λ [ u v ] = i [ i γ Im ( w 2 ) i V i L ^ + γ Re ( w 2 ) L ^ γ Re ( w 2 ) i γ Im ( w 2 ) i V i ] [ u v ] .

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