Abstract

We address the propagation dynamics of gap solitons at the interface between uniform media and an optical lattice in the framework of a nonlinear fractional Schrödinger equation. Different families of solitons residing in the first and second bandgaps of the Floquet-Bloch spectrum are revealed. They feature a combination of the unique properties of fractional diffraction effects, surface waves and gap solitons. The instability of solitons can be remarkably suppressed by the decrease of Lévy index, especially obvious for solitons in the second gaps. Additionally, we study the properties of multi-peaked solitons in fractional dimensions and find that they can be made completely stable in a wide region, provided that their power exceeds a critical value. Counterintuitively, at a small Lévy index, the instability region shrinks with the increase of the number of soliton peaks.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
    [Crossref]
  2. N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
    [Crossref]
  3. N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
    [Crossref]
  4. R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2001).
  5. S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 1117–1120 (2015).
    [Crossref] [PubMed]
  6. B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: The one-dimensional lévy crystal,” Phys. Rev. E 88, 012120 (2013).
    [Crossref]
  7. R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50, 1395–1398 (1983).
    [Crossref]
  8. 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]
  9. A. Kundu and B. Seradjeh, “Transport signatures of Floquet majorana fermions in driven topological superconductors,” Phys. Rev. Lett. 111, 136402 (2013).
    [Crossref] [PubMed]
  10. F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Condens. Matter. 698, 012025 (2016).
  11. J. Lőrinczi and J. Małecki, “Spectral properties of the massless relativistic harmonic oscillator,” J. Differ. Equations 253, 2846–2871 (2012).
    [Crossref]
  12. J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 48, 072105 (2007).
    [Crossref]
  13. B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
    [Crossref]
  14. Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
    [Crossref]
  15. M. Żaba and P. Garbaczewski, “Solving fractional schrödinger-type spectral problems: Cauchy oscillator and Cauchy well,” J. Math. Phys. 55, 092103 (2014).
    [Crossref]
  16. 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]
  17. 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]
  18. 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]
  19. W.-P. Zhong, M. R. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (2016).
    [Crossref]
  20. 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]
  21. C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation with an optical lattice,” Sci. Rep. 7, 5442 (2017).
    [Crossref]
  22. 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] [PubMed]
  23. C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (2016).
    [Crossref] [PubMed]
  24. H. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena (North-Holland, 1991).
  25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006).
    [Crossref] [PubMed]
  26. Y. V. Kartashov, F. Ye, and L. Torner, “Vector mixed-gap surface solitons,” Opt. Express 14, 4808 (2006).
    [Crossref] [PubMed]
  27. W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express 14, 11271 (2006).
    [Crossref] [PubMed]
  28. H. Liu, H. Jin, and L. Dong, “Disordered surface gap solitons,” Phys. Rev. A 89, 023826 (2014).
    [Crossref]
  29. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
    [Crossref]
  30. T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
    [Crossref] [PubMed]
  31. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
    [Crossref]
  32. C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in pt-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (2012).
    [Crossref] [PubMed]

2017 (1)

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

2016 (7)

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] [PubMed]

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

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Condens. Matter. 698, 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]

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]

W.-P. Zhong, M. R. 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]

2015 (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, 180403 (2015).
[Crossref]

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

2014 (2)

M. Żaba and P. Garbaczewski, “Solving fractional schrödinger-type spectral problems: Cauchy oscillator and Cauchy well,” J. Math. Phys. 55, 092103 (2014).
[Crossref]

H. Liu, H. Jin, and L. Dong, “Disordered surface gap solitons,” Phys. Rev. A 89, 023826 (2014).
[Crossref]

2013 (4)

Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (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]

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]

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)

J. Lőrinczi and J. Małecki, “Spectral properties of the massless relativistic harmonic oscillator,” J. Differ. Equations 253, 2846–2871 (2012).
[Crossref]

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

C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in pt-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (2012).
[Crossref] [PubMed]

2009 (1)

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
[Crossref]

2007 (1)

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 48, 072105 (2007).
[Crossref]

2006 (4)

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, F. Ye, and L. Torner, “Vector mixed-gap surface solitons,” Opt. Express 14, 4808 (2006).
[Crossref] [PubMed]

W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express 14, 11271 (2006).
[Crossref] [PubMed]

2002 (1)

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]

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]

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]

Alexander, T. J.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
[Crossref]

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
[Crossref] [PubMed]

Belic, M. R.

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, 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]

W.-P. Zhong, M. R. Belić, 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, 180403 (2015).
[Crossref]

Chen, W. H.

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]

Dong, J.

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 48, 072105 (2007).
[Crossref]

Dong, L.

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] [PubMed]

H. Liu, H. Jin, and L. Dong, “Disordered surface gap solitons,” Phys. Rev. A 89, 023826 (2014).
[Crossref]

C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in pt-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (2012).
[Crossref] [PubMed]

Fan, D.

Garbaczewski, P.

M. Żaba and P. Garbaczewski, “Solving fractional schrödinger-type spectral problems: Cauchy oscillator and Cauchy well,” J. Math. Phys. 55, 092103 (2014).
[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]

He, Y. J.

Herrmann, R.

R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2001).

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]

Jin, H.

H. Liu, H. Jin, and L. Dong, “Disordered surface gap solitons,” Phys. Rev. A 89, 023826 (2014).
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, F. Ye, and L. Torner, “Vector mixed-gap surface solitons,” Opt. Express 14, 4808 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006).
[Crossref] [PubMed]

Kivshar, Y. S.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
[Crossref]

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
[Crossref] [PubMed]

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, Y.

Liu, H.

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.

Lorinczi, J.

J. Lőrinczi and J. Małecki, “Spectral properties of the massless relativistic harmonic oscillator,” J. Differ. Equations 253, 2846–2871 (2012).
[Crossref]

Luchko, Y.

Y. Luchko, “Fractional Schrödinger equation for a particle moving in a potential well,” J. Math. Phys. 54, 012111 (2013).
[Crossref]

Malecki, J.

J. Lőrinczi and J. Małecki, “Spectral properties of the massless relativistic harmonic oscillator,” J. Differ. Equations 253, 2846–2871 (2012).
[Crossref]

Malomed, B. A.

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]

Olivar-Romero, F.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Condens. Matter. 698, 012025 (2016).

Ostrovskaya, E. A.

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, “Self-trapped nonlinear matter waves in periodic potentials,” Phys. Rev. Lett. 96, 040401 (2006).
[Crossref] [PubMed]

Ponath, H. E.

H. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena (North-Holland, 1991).

Rosas-Ortiz, O.

F. Olivar-Romero and O. Rosas-Ortiz, “Factorization of the quantum fractional oscillator,” J. Phys.: Condens. Matter. 698, 012025 (2016).

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]

Stegeman, G. I.

H. E. Ponath and G. I. Stegeman, Nonlinear Surface Electromagnetic Phenomena (North-Holland, 1991).

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]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, F. Ye, and L. Torner, “Vector mixed-gap surface solitons,” Opt. Express 14, 4808 (2006).
[Crossref] [PubMed]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. 96, 073901 (2006).
[Crossref] [PubMed]

Wang, H. Z.

Wang, J.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
[Crossref]

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, 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, 83–130 (2013).
[Crossref]

Xu, C.

Xu, M.

J. Dong and M. Xu, “Some solutions to the space fractional Schrödinger equation using momentum representation method,” J. Math. Phys. 48, 072105 (2007).
[Crossref]

Yang, J.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical lattices,” Phys. Rev. A 79, 043610 (2009).
[Crossref]

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

Ye, F.

Zaba, M.

M. Żaba and P. Garbaczewski, “Solving fractional schrödinger-type spectral problems: Cauchy oscillator and Cauchy well,” J. Math. Phys. 55, 092103 (2014).
[Crossref]

Zhang, L.

Zhang, Y.

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, 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]

W.-P. Zhong, M. R. 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]

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. 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, 83–130 (2013).
[Crossref]

Zhong, H.

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] [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, 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]

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. 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. R. Belić, and Y. Zhang, “Accessible solitons of fractional dimension,” Ann. Phys. 368, 110–116 (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]

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Figures (6)

Fig. 1
Fig. 1 (a, b) Band-gap structure of periodic lattice. p = 4 in (a) and α = 1.2 in (b). Bands are marked with gray color; gaps are shown white. Bandgap spectrum at α = 2 in (c) and 1.2 in (d). p = 4 in (c, d) and Ω = 4 in all the panels.
Fig. 2
Fig. 2 Profiles of surface gap solitons in the first (a–c) and second (d) finite gaps. (a) b = 3.0. (b) b = 2.8. (c) A twisted soliton at b = 4. (d) b = 15, p = 12. p = 4 in (a–c) and α = 1.2 in (b–d).
Fig. 3
Fig. 3 (a) Effective width of solitons versus propagation constant b (top plot) and Lévy index α (bottom). b = 3 in the bottom plot. (b) The peak value of solitons versus b for different α. Dependence of soliton power U versus b in the second gap at α = 2 (c) and 1.2 (d). The lower curves denote single-peaked solitons and the upper ones stand for two-peaked solitons. Solid: stable; dashed: unstable. p = 4 in (a, b) and 12 in (c, d).
Fig. 4
Fig. 4 Profiles of three-peaked solitons at b = 3 (a) and α = 1.2 (b). Profile of five-peaked soliton at b = 3.8 (c) and seven-peaked soliton at b = 3 (d) with α = 1.2. Power U versus b for multi-peaked surface gap solitons at α = 2 (e) and 1.2 (f). Solid: stable; dashed: unstable. All solitons reside in the first gap and p = 4 in all the panels.
Fig. 5
Fig. 5 Linear-stability spectra for the unstable three-peaked and stable seven-peaked solitons marked by open circles in Fig. 4(e) and Fig. 4(f), respectively. b = 0.5 in (a) and 4.7 in (d).
Fig. 6
Fig. 6 Propagation simulations of multi-peaked surface gap solitons. (a, b) Two-peaked solitons in the second gaps of lattices with p = 12 marked in Figs. 3(c) and 3(d). b = 3.8 in (a) and 14.9 in (b). (c–f) Multi-peaked solitons in the first gaps of lattices with p = 4 marked in Figs. 4(e) and 4(f). b = 0.5 in (c, d) and 4.7 in (e, f). α = 2 in (a, c) and 1.2 in other panels. x ∈ [−15, 15] and z = 6000 in (a–f).

Equations (4)

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i q z = 1 2 ( 2 x 2 ) α / 2 q + | q | 2 q R ( x ) q ,
1 2 | k + K n | α C q + m P m C q m = b C q ,
1 2 ( 2 x 2 ) α / 2 w + | w | 2 w R w + b w = 0 ,
λ u = 1 2 ( 2 x 2 ) α / 2 u + w 2 ( 2 u + v ) R u + b u λ v = 1 2 ( 2 x 2 ) α / 2 v w 2 ( 2 v + u ) + R v + b v ,

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