Abstract

We study the solitons in parity-time symmetric potential in the medium with spatially modulated nonlocal nonlinearity. It is found that the coefficient of the spatially modulated nonlinearity and the degree of the uniform nonlocality can profoundly affect the stability of solitons. There exist stable solitons in low-power region, and unstable solitons in high-power region. In the unstable cases, the solitons exhibit jump from the original site to the next one, and they can continue the motion into the other lattices. The region of the stable soliton can be expanded by increasing the coefficient of the modulated nonlocality. Finally, critical amplitude of the imaginary part of the linear PT lattices is obtained, above which solitons are unstable and decay immediately.

©2012 Optical Society of America

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  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]
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    [Crossref] [PubMed]
  3. K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
  4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
  5. C. E. Rüter, K. G. Makris, R. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
  6. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  16. X. Zhu, H. Wang, L. X. Zheng, H. G. 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).
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    [Crossref]
  20. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
    [Crossref]
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    [Crossref] [PubMed]
  22. F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
    [Crossref]
  23. Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
    [Crossref] [PubMed]
  24. 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]
  25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
    [Crossref] [PubMed]
  26. Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  29. G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
    [Crossref]
  30. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
    [Crossref]
  31. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
    [Crossref] [PubMed]
  32. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
    [Crossref]
  33. M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
    [Crossref] [PubMed]

2012 (8)

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

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

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (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]

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

2011 (7)

2010 (5)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

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

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

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

2008 (4)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

2006 (2)

2005 (3)

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

2004 (1)

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

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]

Abdullaev, F. Kh.

F. Kh. 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]

Ablowitz, M. J.

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Akylas, T. I.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

Bender, C. M.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Blömer, D.

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Chen, D.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Chen, L.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

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

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Dreisow, F.

Driben, R.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

EI-Ganainy, R.

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Ganainy, R.

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

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]

Grimm, R.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Guo, Z.

He, Y.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

He, Y. J.

Hecker Denschlag, J.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Hellwig, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Hu, B.

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]

Hwang, G.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

Jiang, X.

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

Kartashov, Y. V.

F. Kh. 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]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Kip, D.

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

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

F. Kh. 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]

Kottos, T.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Li, H.

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

Li, H. G.

Li, L.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Li, R.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Liu, S.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

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

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]

Lu, K.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Ma, C.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (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.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Malomed, B. A.

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

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

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]

Nolte, S.

Ruff, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Rüter, C. E.

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

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Schreiber, T.

Segev, M.

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

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Shi, Z.

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Szameit, A.

Thalhammer, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Theis, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Tünnermann, A.

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Vysloukh, V. A.

Wang, H.

Wang, J.

Winkler, K.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Xu, Z.

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Yang, J.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (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]

Yang, N.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Ye, F.

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

F. Kh. 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, Y.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[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]

Zhou, K.

Zhu, X.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

X. Zhu, H. Wang, L. X. Zheng, H. G. 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] [PubMed]

Nat. Phys. (1)

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

Opt. Commun. (2)

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285(15), 3320–3324 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (8)

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

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

X. Zhu, H. Wang, L. X. Zheng, H. G. 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] [PubMed]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

Phys. Rev. A (8)

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. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (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]

F. Kh. 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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (8)

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

Proc. Romanian Acad. A (1)

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

Stud. Appl. Math. (1)

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

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Figures (10)
Fig. 1
Fig. 1 (a) PT-symmetric optical lattices (OLs) with W0 = 2 (the bue line is the real part, and the red line is the imaginary part). (b)The band structure corresponding to the lattice profile shown in panel (a).

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2 Power P versus propagation constant μ for various k with d = 1 and W0 = 2.The solid line represents the stable regions and the dash line represents the unstable regions.

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Fig. 3
Fig. 3 Soliton profiles (blue line is the real part, the red line is the imaginary part, and the green dash line is the refractive index profile) and Soliton evolution for μ = 1.8 (a, c), and μ = 2.5 (b, d). The other parameters are d = 1, W0 = 2, and k = −0.8.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4 The soliton profiles for μ = 2.0 (a), and μ = 2.8 (b). (c) and (d) are the corresponding propagations. Here, k = 0. The other parameters are the same as those in Fig. 3.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5 The soliton profiles for μ = 2.2 (a), and μ = 3.0 (b). (c), (d) are the corresponding propagations. Here, k = 1. The other parameters are the same as those in Fig. 3.

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6 When μ = 2.0. (a). (b), (c) are the soliton profiles for k = −0.8, k = 0, and k = 1, respectively.

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Fig. 7
Fig. 7 Power P versus propagation constant μ (a). The soliton profiles for μ = 1.9 (b), and μ = 2.5 (c). (d), (e) are the corresponding propagations. Here, d = 3. The other parameters are the same as those in Fig. 5.

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Fig. 8
Fig. 8 Stability domain (μ, k) for (a) W0 = 1.2, (b) W0 = 2, and (c) W0 = 2.4 when d = 1. (d) stability domain (μ, d) for W0 = 2 and k = 1. The gray regions are stable domains.

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Fig. 9
Fig. 9 (a) PT symmetric OLs with W0 = 4.2 (the blue line is the real part, and the red line is the imaginary part). (b) The band structure corresponding to the lattice profile shown in panel (a).

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Fig. 10
Fig. 10 The soliton profiles for μ = 1.0 (a) and μ = 2.0 (b), and. (c) and (d) are their corresponding propagations. The parameters are d = 1, W0 = 4.2, and k = 1.

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Equations (5)

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

i ∂q ∂z + 1 2 ∂ 2 q ∂ x 2 +(V+iW)q+[1+f(x)]q ∫ −∞ +∞ g(x−λ) | q(λ) | 2 dλ=0,
g(x)=1/(2 d 1 2 )exp(−| x |/ d 1 2 )
1 2 ∂ 2 u ∂ x 2 +[V(x)+iW(x)]u+[1+f(x)]u ∫ −∞ +∞ g(x−λ) | u(λ) | 2 dλ−μu=0.
1 2 ∂ 2 u ∂ x 2 +[V(x)+iW(x)]u=μu.
1 2 ( ∂ 2 ∂ x 2 +2iK ∂ ∂x − K 2 ) F K +[V(x)+iW(x)] F K =μ F K .

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