Abstract

We investigate new types of gap solitons in a periodic parity-time (PT)-symmetric lattice with fractional-order diffraction. Both the fundamental and dipole solitons in the first and second gaps are discussed. It is found that fractional-order diffraction can not only stabilize low-power dipole PT solitons in the first gap under focusing nonlinearity, but also help to get stable dipole PT solitons in the second gap under defocusing nonlinearity. Additionally, the influence of the strength ${w_i}$ of the gain–loss component on the properties of solitons is also analyzed. It is shown that increasing ${w_i}$ is unfavorable to the stability of fractional fundamental solitons, especially for the second gap, while for fractional dipole solitons, the increase of ${w_i}$ may lead to their destabilization in the first gap, but stabilization in the second gap.

© 2020 Optical Society of America

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References

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  1. R. B. Laughlin, “Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
    [Crossref]
  2. R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339, 1–77 (2000).
    [Crossref]
  3. L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
    [Crossref]
  4. J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5, 83–130 (2013).
    [Crossref]
  5. N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
    [Crossref]
  6. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
    [Crossref]
  7. N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
    [Crossref]
  8. B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88, 012120 (2013).
    [Crossref]
  9. S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 1117–1120 (2015).
    [Crossref]
  10. Y. Zhang, X. Liu, M. R. Belic, 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]
  11. 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]
  12. C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation with an optical lattice,” Sci. Rep. 7, 5442 (2017).
    [Crossref]
  13. Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
    [Crossref]
  14. B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
    [Crossref]
  15. 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, 14406–14418 (2016).
    [Crossref]
  16. S. Duo and Y. Zhang, “Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation,” Comput. Math. Appl. 71, 2257–2271 (2016).
    [Crossref]
  17. 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]
  18. C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (2016).
    [Crossref]
  19. J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
    [Crossref]
  20. Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
    [Crossref]
  21. X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43, 5749–5752 (2018).
    [Crossref]
  22. M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
    [Crossref]
  23. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [Crossref]
  24. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [Crossref]
  25. C. E. Ruter, 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]
  26. 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]
  27. S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
    [Crossref]
  28. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
    [Crossref]
  29. 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]
  30. L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
    [Crossref]
  31. 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, 526–531 (2016).
    [Crossref]
  32. L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a PT-symmetric potential,” Opt. Express 26, 10509–10518 (2018).
    [Crossref]
  33. C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
    [Crossref]
  34. X. Yao and X. Liu, “Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential,” Photon. Res. 6, 875–879 (2018).
    [Crossref]
  35. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
    [Crossref]
  36. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
    [Crossref]
  37. T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
    [Crossref]
  38. S. K. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” Europhys. Lett. 79, 50003 (2007).
    [Crossref]

2018 (7)

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref]

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43, 5749–5752 (2018).
[Crossref]

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

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

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

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

2017 (3)

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

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

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]

2016 (6)

C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (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, 14406–14418 (2016).
[Crossref]

S. Duo and Y. Zhang, “Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation,” Comput. Math. Appl. 71, 2257–2271 (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]

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (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, 526–531 (2016).
[Crossref]

2015 (2)

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

Y. Zhang, X. Liu, M. R. Belic, 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]

2013 (2)

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

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

2012 (2)

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

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
[Crossref]

2011 (1)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

2010 (1)

C. E. Ruter, 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)

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

2008 (2)

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]

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]

2007 (3)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

S. K. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” Europhys. Lett. 79, 50003 (2007).
[Crossref]

2006 (1)

T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
[Crossref]

2002 (1)

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

2000 (3)

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

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

R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339, 1–77 (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]

Adhikari, S. K.

S. K. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” Europhys. Lett. 79, 50003 (2007).
[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]

Belic, M. R.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (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, 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, 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, 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 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]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Chen, M.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[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, 526–531 (2016).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

C. E. Ruter, 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]

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]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (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]

Deng, H.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Dong, L.

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

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref]

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation with an optical lattice,” 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, 5636–5639 (2016).
[Crossref]

Duo, S.

S. Duo and Y. Zhang, “Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation,” Comput. Math. Appl. 71, 2257–2271 (2016).
[Crossref]

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

El-Ganainy, R.

C. E. Ruter, 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]

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]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

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, 14406–14418 (2016).
[Crossref]

Furdyna, J. K.

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
[Crossref]

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, Q.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

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]

Hu, W.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[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.

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26, 2650–2658 (2018).
[Crossref]

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

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation with an optical lattice,” 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, 5636–5639 (2016).
[Crossref]

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]

Kip, D.

C. E. Ruter, 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]

Klafter, J.

R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339, 1–77 (2000).
[Crossref]

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Lai, T.

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
[Crossref]

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

Laskin, N.

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

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

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

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, 14406–14418 (2016).
[Crossref]

Li, C.

Li, H.

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
[Crossref]

Li, J.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Li, L.

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
[Crossref]

Li, 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]

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, 14406–14418 (2016).
[Crossref]

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Liu, X.

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

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43, 5749–5752 (2018).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, 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]

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
[Crossref]

Longhi, S.

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

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Lu, D.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Makris, K. G.

C. E. Ruter, 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]

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]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Malomed, B. A.

S. K. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” Europhys. Lett. 79, 50003 (2007).
[Crossref]

T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
[Crossref]

Mayteevarunyoo, T.

T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
[Crossref]

Metzler, R.

R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339, 1–77 (2000).
[Crossref]

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]

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]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[Crossref]

Rokhinson, L. P.

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
[Crossref]

Ruter, C. E.

C. E. Ruter, 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]

Segev, M.

C. E. Ruter, 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]

Stickler, B. A.

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

Tian, Z.

Wang, Q.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Wang, R.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

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]

Wen, J.

Xiao, 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, 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, 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, 83–130 (2013).
[Crossref]

Xie, W.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

Xu, C.

Yang, J.

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

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

Yao, X.

Ye, F.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Zeng, S.

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Zhang, J.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Zhang, L.

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[Crossref]

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, 14406–14418 (2016).
[Crossref]

Zhang, W.

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

Zhang, Y.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (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, 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]

S. Duo and Y. Zhang, “Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation,” Comput. Math. Appl. 71, 2257–2271 (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, 526–531 (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, 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, 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. Belic, 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, 83–130 (2013).
[Crossref]

Zhong, H.

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (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, 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, 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 Photon. Rev. 10, 526–531 (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, 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belic, 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]

Zhu, X.

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
[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, 526–531 (2016).
[Crossref]

Adv. Opt. Photon. (1)

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]

Comput. Math. Appl. (1)

S. Duo and Y. Zhang, “Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation,” Comput. Math. Appl. 71, 2257–2271 (2016).
[Crossref]

Europhys. Lett. (3)

C. Huang, H. Deng, W. Zhang, F. Ye, and L. Dong, “Fundamental solitons in the nonlinear fractional Schrödinger equation with a PT-symmetric potential,” Europhys. Lett. 122, 24002 (2018).
[Crossref]

S. K. Adhikari and B. A. Malomed, “Tightly bound gap solitons in a Fermi gas,” Europhys. Lett. 79, 50003 (2007).
[Crossref]

Q. Wang, J. Li, L. Zhang, and W. Xie, “Hermite-Gaussian-like soliton in the nonlocal nonlinear fractional Schrödinger equation,” Europhys. Lett. 122, 64001 (2018).
[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. Opt. (1)

L. Li, X. Zhu, H. Li, and T. Lai, “Vector solitons in parity-time symmetric lattices with nonlocal nonlinearity,” J. Opt. 18, 095501 (2016).
[Crossref]

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, 526–531 (2016).
[Crossref]

Nat. Phys. (2)

C. E. Ruter, 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]

L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c. Josephson effect in a semiconductor-superconductor nanowire as a signature of Majorana particles,” Nat. Phys. 8, 795–799 (2012).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Photon. Res. (1)

Phys. Lett. A (1)

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

Phys. Rep. (1)

R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Phys. Rep. 339, 1–77 (2000).
[Crossref]

Phys. Rev. A (1)

T. Mayteevarunyoo and B. A. Malomed, “Stability limits for gap solitons in a Bose-Einstein condensate trapped in a time-modulated optical lattice,” Phys. Rev. A 74, 033616 (2006).
[Crossref]

Phys. Rev. E (4)

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]

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

M. Chen, S. Zeng, D. Lu, W. Hu, and Q. Guo, “Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity,” Phys. Rev. E 98, 022211 (2018).
[Crossref]

Phys. Rev. Lett. (7)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
[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]

S. Longhi, “Bloch oscillations in complex crystals with PT symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106, 213901 (2011).
[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]

Y. Zhang, X. Liu, M. R. Belic, 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]

R. B. Laughlin, “Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
[Crossref]

Sci. Rep. (3)

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 with an optical lattice,” Sci. Rep. 7, 5442 (2017).
[Crossref]

Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation,” Sci. Rep. 7, 17872 (2017).
[Crossref]

Stud. Appl. Math. (1)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

Other (1)

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

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the PT-symmetric lattice. (Blue solid and red dashed lines represent the real and imaginary parts of the PT-symmetric potential, respectively. White fringes denote ${\rm Re}\,V(x)\; \gt {4}$, while gray ones ${\rm Re}\,V(x)\; \le \;{4}$.) (b) The Bloch band structure versus Lévy index $\alpha $. Black filled regions are Bloch bands. (c) The band structure of the lattice with $\alpha = {2}$. (d) The band structure of the lattice with $\alpha = {1.2}$. In (a)–(d), ${w_i} = {0.1}$. (e) The band structure of the lattice with ${w_i} = {0.5}$, $\alpha = {1.2}$. (f) The band structure of the lattice with ${w_i} = {0.51}$, $\alpha = {1.2}$.
Fig. 2.
Fig. 2. Power $P$ versus propagation constant $\mu $ for solitons in the first gap at (a) $\alpha = {1.2}$, (b) $\alpha = {2}$ (the gray region is a Bloch band; red curves, dipole solitons; blue curves, fundamental solitons; solid curves, stable; dashed curves, unstable). In all cases, ${w_i} = {0.1}$.
Fig. 3.
Fig. 3. Profile of dipole soliton in the first gap at $\mu = {3.5}$ for (a) $\alpha = {2}$, (b) $\alpha = {1.2}$ (the blue solid and red dashed curves are the real and imaginary parts, respectively). (c) and (d) Corresponding propagation of solitons in (a) and (b). In all cases, $\sigma = {1}$, ${w_i} = {0.1}$.
Fig. 4.
Fig. 4. Power $P$ versus propagation constant $\mu $ for solitons in the second gap. (a) Fundamental soliton; (b) dipole soliton (the solid and dashed lines represent the stable and unstable ranges). (c) Profile of the fundamental soliton for $\mu = {2}$ (the blue solid and red dashed curves are the real and imaginary parts, respectively). (d) The corresponding linear-stability spectra; (e) corresponding unstable propagation; (f) profile of the dipole soliton for $\mu = {2.4}$; (g) corresponding linear-stability spectra; (h) corresponding stable propagation. In all cases, $\sigma = - {1}$, $\alpha = {1.2},\;{w_i} = {0.1}$.
Fig. 5.
Fig. 5. Power of solitons versus ${w_i}$. (a) Fundamental solitons in the first gap, $\mu = {4},\;\sigma = - {1}$; (b) fundamental solitons in the second gap, $\mu = {2}$, $\sigma = - {1}$ (the solid and dashed lines represent the stable and unstable ranges); (c) profile of the fundamental soliton for $\mu = {4}$, $\sigma = - {1}$, ${w_i} = {0.32}$ (the blue solid and red dashed curves are the real and imaginary parts, respectively); (d) corresponding linear-stability spectra; (e) corresponding unstable propagation; (f) profile of the fundamental soliton for $\mu = {2}$, $\sigma = - {1}$, ${w_i} = {0.01}$; (g) corresponding linear-stability spectra; (h) corresponding stable propagation. In all cases, $\alpha = {1.2}$.
Fig. 6.
Fig. 6. Power of solitons versus ${w_i}$. (a) Dipole solitons in the first gap, $\mu = {3.5}$, $\sigma = {1}$; (b) dipole solitons in the second gap, $\mu = {2.2}$, $\sigma = - {1}$ (the solid and dashed lines represent the stable and unstable ranges); (c) profile of the dipole soliton for $\mu = {3.5}$, $\sigma = {1}$, ${w_i} = {0.15}$ (the blue solid and red dashed curves are the real and imaginary parts, respectively); (d) corresponding linear-stability spectra; (e) corresponding unstable propagation; (f) profile of the dipole soliton for $\mu = {2.2},\;\sigma = - {1}$, ${w_i} = {0.2}$; (g) corresponding linear-stability spectra; (h) corresponding stable propagation. In all cases, $\alpha = {1.2}$.

Equations (9)

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

i U z ( 2 x 2 ) α / 2 U + V ( x ) U + σ | U | 2 U = 0 ,
V ( x ) = V 0 ( cos 2 ( x ) + i w i sin ( 2 x ) ) ,
U ( x , z ) = f ( x ) exp ( i μ z ) ,
( 2 x 2 ) α / 2 f + V ( x ) f + σ | f | 2 f μ f = 0 ,
( 2 x 2 ) α / 2 f + V ( x ) f μ f = 0.
n [ μ | k + K n | α ] C n exp [ i ( k + K n ) x ] + m , n P m C n exp [ i ( k + K n + K m ) x ] = 0.
| k + K q | α C q + m P m C q m = μ C q .
q ( x , z ) = exp ( i μ z ) { f ( x ) + [ F ( x ) exp ( δ z ) + G ( x ) exp ( δ z ) ] } ,
{ δ F = i [ ( 2 x 2 ) α / 2 F μ F + V F + 2 | f | 2 F + f 2 G ] δ G = i [ ( 2 x 2 ) α / 2 G + μ G V G 2 | f | 2 G f 2 F ] .