Abstract

We investigate the existence and stability of in-phase three-pole and four-pole gap solitons in the fractional Schrödinger equation supported by one-dimensional parity-time-symmetric periodic potentials (optical lattices) with defocusing Kerr nonlinearity. These solitons exist in the first finite gap and are stable in the moderate power region. When the Lévy index decreases, the stable regions of these in-phase multipole gap solitons shrink. Below a Lévy index threshold, the effective multipole soliton widths decrease as the Lévy index increases. Above the threshold, these solitons become less localized as the Lévy index increases. The Lévy index cannot change the phase transition point of the PT-symmetric optical lattices. We also study transverse power flow in these multipole gap solitons.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  4. S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40(6), 1117–1120 (2015).
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  5. Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115(18), 180403 (2015).
    [Crossref]
  6. Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photonics Rev. 10(3), 526–531 (2016).
    [Crossref]
  7. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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    [Crossref]
  12. X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43(23), 5749–5752 (2018).
    [Crossref]
  13. L. Dong and Z. Tian, “Truncated-Bloch-wave solitons in nonlinear fractional periodic systems,” Ann. Phys. 404, 57–65 (2019).
    [Crossref]
  14. L. Zeng and J. Zeng, “One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice,” Opt. Lett. 44(11), 2661–2664 (2019).
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    [Crossref]
  16. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
    [Crossref]
  17. Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87(1), 013816 (2013).
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  18. S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35(17), 2976–2978 (2010).
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  21. 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(1), 013831 (2012).
    [Crossref]
  22. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
    [Crossref]
  23. H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
    [Crossref]
  24. 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(1), 013812 (2014).
    [Crossref]
  25. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
    [Crossref]
  26. Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38(14), 2600–2603 (2013).
    [Crossref]
  27. M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
    [Crossref]
  28. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
    [Crossref]
  29. C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20(15), 16823–16831 (2012).
    [Crossref]
  30. C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37(21), 4543–4545 (2012).
    [Crossref]
  31. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
    [Crossref]
  32. X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
    [Crossref]
  33. X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013).
    [Crossref]
  34. 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(2), 24002 (2018).
    [Crossref]
  35. X. Yao and X. Liu, “Solitons in the fractional Schrödinger equation with parity-time-symmetric lattice potential,” Photonics Res. 6(9), 875–879 (2018).
    [Crossref]
  36. L. Dong and C. Huang, “Double-hump solitons in fractional dimensions with a PT-symmetric potential,” Opt. Express 26(8), 10509–10518 (2018).
    [Crossref]
  37. J. Xie, X. Zhu, and Y. He, “Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices,” Nonlinear Dyn. 97(2), 1287–1294 (2019).
    [Crossref]
  38. C. Huang and L. Dong, “Dissipative surface solitons in a nonlinear fractional Schrödinger equation,” Opt. Lett. 44(22), 5438–5441 (2019).
    [Crossref]
  39. L. Dong, C. Huang, and W. Qi, “Nonlocal solitons in fractional dimensions,” Opt. Lett. 44(20), 4917–4920 (2019).
    [Crossref]
  40. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
    [Crossref]
  41. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Physik 52(7-8), 555–600 (1929).
    [Crossref]
  42. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
  43. C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40(12), 2758–2761 (2015).
    [Crossref]

2019 (5)

2018 (5)

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26(3), 2650–2658 (2018).
[Crossref]

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43(23), 5749–5752 (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(2), 24002 (2018).
[Crossref]

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

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

2016 (3)

2015 (4)

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

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

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40(12), 2758–2761 (2015).
[Crossref]

2014 (1)

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(1), 013812 (2014).
[Crossref]

2013 (5)

2012 (7)

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20(15), 16823–16831 (2012).
[Crossref]

C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37(21), 4543–4545 (2012).
[Crossref]

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (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(1), 013831 (2012).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

2011 (3)

2010 (1)

2008 (1)

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

2007 (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(2), 153–197 (2007).
[Crossref]

2002 (1)

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

2000 (2)

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

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

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Physik 52(7-8), 555–600 (1929).
[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(4), 041805 (2011).
[Crossref]

Achilleos, V.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

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(1), 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(1), 013812 (2014).
[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 Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

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

Bender, C. M.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bersch, C.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Physik 52(7-8), 555–600 (1929).
[Crossref]

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87(1), 013816 (2013).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 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(1), 013812 (2014).
[Crossref]

Carretero-González, R.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

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

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[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(2), 24002 (2018).
[Crossref]

Dmitriev, S. V.

Dong, L.

L. Dong and Z. Tian, “Truncated-Bloch-wave solitons in nonlinear fractional periodic systems,” Ann. Phys. 404, 57–65 (2019).
[Crossref]

L. Dong, C. Huang, and W. Qi, “Nonlocal solitons in fractional dimensions,” Opt. Lett. 44(20), 4917–4920 (2019).
[Crossref]

C. Huang and L. Dong, “Dissipative surface solitons in a nonlinear fractional Schrödinger equation,” Opt. Lett. 44(22), 5438–5441 (2019).
[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(2), 24002 (2018).
[Crossref]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26(3), 2650–2658 (2018).
[Crossref]

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

C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT symmetry,” Opt. Express 21(3), 3917–3925 (2013).
[Crossref]

C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37(21), 4543–4545 (2012).
[Crossref]

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20(15), 16823–16831 (2012).
[Crossref]

Driben, R.

EI-Ganainy, R.

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

Fan, D.

Frantzeskakis, D. J.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

Hang, C.

He, W.

He, Y.

J. Xie, X. Zhu, and Y. He, “Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices,” Nonlinear Dyn. 97(2), 1287–1294 (2019).
[Crossref]

X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013).
[Crossref]

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
[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(1), 013831 (2012).
[Crossref]

He, Y.-J.

Hu, S.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Hu, W.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Huang, C.

Huang, G.

Jiang, X.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[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(1), 013812 (2014).
[Crossref]

Kartashov, Y. V.

Kevrekidis, P. G.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Kivshar, Y. S.

Konotop, V. V.

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40(12), 2758–2761 (2015).
[Crossref]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87(1), 013816 (2013).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[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(2), 153–197 (2007).
[Crossref]

Laskin, N.

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

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

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

Lei, D.

Li, C.

Li, H.

Li, Y.

Liu, H.

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(1), 013831 (2012).
[Crossref]

Liu, X.

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

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43(23), 5749–5752 (2018).
[Crossref]

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

Longhi, S.

Lu, D.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Ma, X.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Makris, K. G.

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

Malomed, B. A.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87(1), 013816 (2013).
[Crossref]

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
[Crossref]

Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

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(1), 013831 (2012).
[Crossref]

Miri, M.-A.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

Nixon, S.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

Peschel, U.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Qi, W.

Regensburger, A.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Shi, Z.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

Sukhorukov, A. A.

Tian, Z.

L. Dong and Z. Tian, “Truncated-Bloch-wave solitons in nonlinear fractional periodic systems,” Ann. Phys. 404, 57–65 (2019).
[Crossref]

J. Xiao, Z. Tian, C. Huang, and L. Dong, “Surface gap solitons in a nonlinear fractional Schrödinger equation,” Opt. Express 26(3), 2650–2658 (2018).
[Crossref]

Wang, H.

Wimmer, M.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Xiao, J.

Xiao, M.

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

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

Xie, J.

J. Xie, X. Zhu, and Y. He, “Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices,” Nonlinear Dyn. 97(2), 1287–1294 (2019).
[Crossref]

Xu, C.

Yang, J.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[Crossref]

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

Yao, X.

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

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43(23), 5749–5752 (2018).
[Crossref]

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(2), 24002 (2018).
[Crossref]

Zeng, J.

Zeng, L.

Zezyulin, D. A.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[Crossref]

Zhang, L.

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(2), 24002 (2018).
[Crossref]

Zhang, Y.

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

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

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

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

Zheng, L.-X.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[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 Photonics 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]

Zhong, W.

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

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

Zhu, X.

J. Xie, X. Zhu, and Y. He, “Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices,” Nonlinear Dyn. 97(2), 1287–1294 (2019).
[Crossref]

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
[Crossref]

X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013).
[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(1), 013831 (2012).
[Crossref]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (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 Photonics Rev. 10(3), 526–531 (2016).
[Crossref]

Ann. Phys. (1)

L. Dong and Z. Tian, “Truncated-Bloch-wave solitons in nonlinear fractional periodic systems,” Ann. Phys. 404, 57–65 (2019).
[Crossref]

Europhys. Lett. (1)

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(2), 24002 (2018).
[Crossref]

J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

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

Nat. Commun. (1)

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref]

Nonlinear Dyn. (1)

J. Xie, X. Zhu, and Y. He, “Vector solitons in nonlinear fractional Schrödinger equations with parity-time-symmetric optical lattices,” Nonlinear Dyn. 97(2), 1287–1294 (2019).
[Crossref]

Opt. Express (5)

Opt. Lett. (13)

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]

C. Li, C. Huang, H. Liu, and L. Dong, “Multipeaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37(21), 4543–4545 (2012).
[Crossref]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35(17), 2976–2978 (2010).
[Crossref]

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
[Crossref]

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
[Crossref]

X. Yao and X. Liu, “Off-site and on-site vortex solitons in space-fractional photonic lattices,” Opt. Lett. 43(23), 5749–5752 (2018).
[Crossref]

L. Zeng and J. Zeng, “One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice,” Opt. Lett. 44(11), 2661–2664 (2019).
[Crossref]

L. Dong, C. Huang, and W. Qi, “Nonlocal solitons in fractional dimensions,” Opt. Lett. 44(20), 4917–4920 (2019).
[Crossref]

C. Huang and L. Dong, “Dissipative surface solitons in a nonlinear fractional Schrödinger equation,” Opt. Lett. 44(22), 5438–5441 (2019).
[Crossref]

Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38(14), 2600–2603 (2013).
[Crossref]

X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013).
[Crossref]

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

C. Hang, Y. V. Kartashov, G. Huang, and V. V. Konotop, “Localization of light in a parity-time-symmetric quasi-periodic lattice,” Opt. Lett. 40(12), 2758–2761 (2015).
[Crossref]

Photonics Res. (1)

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

Phys. Lett. A (1)

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

Phys. Rev. A (8)

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87(1), 013816 (2013).
[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(1), 013831 (2012).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

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(1), 013812 (2014).
[Crossref]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[Crossref]

Phys. Rev. E (2)

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

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

Phys. Rev. Lett. (3)

Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

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

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(2), 153–197 (2007).
[Crossref]

Z. Physik (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Physik 52(7-8), 555–600 (1929).
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Other (1)

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

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Figures (6)

Fig. 1.
Fig. 1. (a) is the band structure for α=1.6 and W0=0.1. (b) and (c) are the real and imaginary parts of the band structure when α=1.6 and W0=0.55. (d) is the band structure for α=1.3 and W0=0.1. The real and imaginary parts of the band structure when α=1.3 and W0=0.55 are shown in (e) and (f), respectively.

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Fig. 2.
Fig. 2. (a) Power diagram of the in-phase three-pole solitons (blue solid and red dashed lines represent the stable and unstable cases, respectively, and the shaded regions are the Bloch bands). (b), (c), and (d) are the in-phase three-pole soliton profile, the transverse power flow density within the soliton, and the linear-stability spectrum of the soliton when µ=1.8, respectively. (e) shows the corresponding stable propagation of the perturbed soliton. (f) and (g) are the linear-stability spectrum of the soliton and the unstable propagation of the perturbed in-phase three-pole soliton for µ=1.15, respectively. The linear-stability spectrum of the in-phase three-pole soliton and the unstable propagation of the perturbed soliton when µ=3.5 are illustrated in (h) and (i), respectively. The other parameters are W0=0.1 and α=1.6

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Fig. 3.
Fig. 3. Here, W0=0.1 and α=1.3. (a) Soliton power of in-phase three-pole solitons versus propagation constant. The profile and the linear-stability spectrum of the in-phase three-pole soliton when µ=2.5 are shown in (b) and (c), respectively. (d) is the corresponding stable propagation of the perturbed soliton. (e) and (f) depict the profile and the linear-stability spectrum of the in-phase three-pole soliton when µ=1.8. (g) shows the corresponding unstable propagation of the perturbed soliton. The linear-stability spectrum of the in-phase three-pole soliton and the unstable propagation of the perturbed soliton when µ=3.6 are shown in (h) and (i), respectively.

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Fig. 4.
Fig. 4. (a) Power diagram of the in-phase four-pole solitons. (b), (c), and (d) are the profile, the transverse power flow density, and the linear-stability spectrum of the in-phase four-pole soliton for µ=1.8, respectively. (e) shows the corresponding stable propagation of the perturbed soliton. The linear-stability spectra of the two in-phase four-pole solitons when µ=1.15 and µ=3.6 are shown in (f) and (h), respectively. (g) and (i) are the corresponding unstable propagations of the two perturbed solitons. The other parameters are W0=0.1 and α=1.6.

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Fig. 5.
Fig. 5. Here, W0=0.1 and α=1.3. (a) Soliton power of the in-phase four-pole soliton versus the propagation constant. (b) and (c) show the profile and the linear-stability spectrum of the in-phase four-pole soliton when µ=2.5. (d) is the corresponding stable propagation of the perturbed soliton. The profile and the linear-stability spectrum of the in-phase four-pole soliton for µ=1.8 are shown in (e) and (f), respectively. (g) depicts the corresponding unstable propagation of the perturbed soliton. (h) is the linear-stability spectrum of the in-phase four-pole soliton when µ=3.6. (i) shows the corresponding unstable propagation of the perturbed soliton.

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Fig. 6.
Fig. 6. (a) and (b) are the stability regions (gray domains) of the three-pole and four-pole solitons versus the Lévy index. (c) and (d) are form factors of the in-phase three-pole and four-pole solitons for µ=1.8, respectively. The solid [1.49≤α≤2 in (c) and 1.5≤α≤2 in (d)] and dashed lines represent stable and unstable cases. The other parameter is W0=0.1.

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Equations (6)

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

i ∂ U ∂ z − ( − ∂ 2 ∂ x 2 ) α / α 2 2 U + V ( x ) U − | U | 2 U = 0.
− ( − ∂ 2 ∂ x 2 ) α / α 2 2 q + V ( x ) q − | q | 2 q − μ q = 0.
− ( − ∂ 2 ∂ x 2 ) α / α 2 2 q + V ( x ) q = μ q .
− | k + K q | α C q + ∑ m D m C q − m = μ C q .
U ( x , z ) = e i μ z [ q ( x ) + g ( x ) e δ z + t ∗ ( x ) e δ ∗ z ] .
{ δ g = { [ − μ − ( − ∂ 2 ∂ x 2 ) α / α 2 2 + V − 2 | q | 2 ] g − q 2 t } , δ t = { ( q 2 ) ∗ g + [ μ + ( − ∂ 2 ∂ x 2 ) α / α 2 2 − V ∗ + 2 | q | 2 ] t } .

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