Abstract

We address the existence and stability properties of optical solitons in a competing cubic-quintic medium with an imprinted complex lattice featuring a parity-time (𝒫𝒯) symmetry. Various families of solitons with even and odd geometrical symmetries are found in both the semi-infinite and the first finite gaps. Linear stability analysis corroborated by direct propagation simulations reveals that solitons with different symmetries and different number of humps can propagate stably at the same propagation constants, i.e., multi-stable solitons can exist in this scheme. Interestingly enough, in sharp contrast to the stability of solitons in a conventional (real) lattice, both even and odd solitons with the same propagation constant belonging to different branches can be stable in the first gap of 𝒫𝒯 lattice, which indicates that the imaginary part of lattice plays an important role for the stabilization of solitons.

© 2012 OSA

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  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett.80, 5243–5246 (1998).
    [CrossRef]
  2. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
    [CrossRef]
  3. C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
    [CrossRef]
  4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
    [CrossRef] [PubMed]
  5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef] [PubMed]
  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, 093902 (2009).
    [CrossRef] [PubMed]
  7. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef] [PubMed]
  9. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
    [CrossRef]
  10. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  11. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯 -symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
    [CrossRef]
  12. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
    [CrossRef]
  13. 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, 3320–3324 (2012).
    [CrossRef]
  14. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in 𝒫𝒯 -symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
    [CrossRef]
  15. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
    [CrossRef] [PubMed]
  16. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).
    [CrossRef] [PubMed]
  17. Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).
    [CrossRef] [PubMed]
  18. 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, 043826 (2012).
    [CrossRef]
  19. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with 𝒫𝒯 -symmetric potentials,” Phys. Rev. A 84, 053855 (2011).
    [CrossRef]
  20. S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. 285, 1934–1939 (2012).
    [CrossRef]
  21. 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).
  22. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 023822 (2012).
    [CrossRef]
  23. A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
    [CrossRef] [PubMed]
  24. J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
    [CrossRef]
  25. 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]

2012 (6)

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, 3320–3324 (2012).
[CrossRef]

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

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

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. A8 5, 023822 (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, 043826 (2012).
[CrossRef]

2011 (6)

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

H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).
[CrossRef] [PubMed]

Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).
[CrossRef] [PubMed]

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[CrossRef]

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

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
[CrossRef]

2010 (3)

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

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

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

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, 093902 (2009).
[CrossRef] [PubMed]

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]

2008 (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

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

2007 (1)

2005 (1)

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

1999 (1)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
[CrossRef]

1998 (1)

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

1985 (1)

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[CrossRef] [PubMed]

Abdullaev, F. K.

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

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

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]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
[CrossRef]

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

Boettcher, S.

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

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Cai, T.

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[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, “Solitons in PT-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[CrossRef]

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

Christodoulides, D.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[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, 063807 (2010).
[CrossRef]

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

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

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

Dong, L.

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Dunne, G. V.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
[CrossRef]

El-Ganainy, R.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[CrossRef]

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

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

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

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 023822 (2012).
[CrossRef]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Guo, Z.

He, Y.

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

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, 3320–3324 (2012).
[CrossRef]

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, 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, 043826 (2012).
[CrossRef]

Jiang, X.

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

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[CrossRef] [PubMed]

Kartashov, Y. V.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
[CrossRef]

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

Kip, D.

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

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]

Konotop, V. V.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
[CrossRef]

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

Li, H.

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

Li, 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).

Li, Y.-P.

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

Liu, J.

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

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, 3320–3324 (2012).
[CrossRef]

Liu, S.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. 285, 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, 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, 043826 (2012).
[CrossRef]

Lu, K.

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

Lu, Z.

Ma, C.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in symmetric lattices with competing nonlinearity,” Opt. Commun. 285, 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, 043826 (2012).
[CrossRef]

Makris, K.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[CrossRef]

Makris, K. G.

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

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

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

Meisinger, P. N.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
[CrossRef]

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, 3320–3324 (2012).
[CrossRef]

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

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Musslimani, Z.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[CrossRef]

Musslimani, Z. H.

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

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

Nixon, S.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 023822 (2012).
[CrossRef]

Rüter, C. E.

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

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Segev, M.

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

Shi, Z.

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

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

Wang, H.

Wang, J.

H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).
[CrossRef] [PubMed]

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

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. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

Yang, J.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 023822 (2012).
[CrossRef]

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]

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.

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

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, 041805 (2011).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Zhang, Y.

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

Zhang, Z.-M.

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, 043826 (2012).
[CrossRef]

Zhou, K.

Zhu, X.

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

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, 3320–3324 (2012).
[CrossRef]

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

EPL-Europhys. Lett. (1)

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” EPL-Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Int. J. of Theor. Phys. (1)

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. of Theor. Phys. 50, 1019–1041 (2011).
[CrossRef]

Nat. Phys. (1)

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

Opt. Commun. (2)

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, 3320–3324 (2012).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (2)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, “Complex periodic potentials with real band spectra,” Phys. Lett. A 252, 272–276 (1999).
[CrossRef]

J. Wang, F. Ye, L. Dong, T. Cai, and Y.-P. Li, “Lattice solitons supported by competing cubic-quintic nonlinearity,” Phys. Lett. A 339, 74–82 (2005).
[CrossRef]

Phys. Rev. A (7)

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]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. A8 5, 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, 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, 043826 (2012).
[CrossRef]

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

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

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

Phys. Rev. Lett. (6)

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef] [PubMed]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯 -symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef] [PubMed]

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

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[CrossRef]

A. E. Kaplan, “Bistable solitons,” Phys. Rev. Lett. 55, 1291–1294 (1985).
[CrossRef] [PubMed]

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).

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Figures (8)

Fig. 1
Fig. 1

(a) Typical profile of ���� lattice. Solid: real part; dashed: imaginary part. (b) Band-gap structure of lattices. Solid: V0 = 3, W0 = 0.3 [corresponding to (a)]; dashed: V0 = 3, W0 = 1.55. Inset: imaginary part of complex eigenvalues at W0 = 1.55.

Fig. 2
Fig. 2

(a) Power P of even solitons versus propagation constant b in the semi-infinite gap. The first five branches are plotted. Shaded region denotes the first band. (b–f) Profiles of solitons marked by dots in (a) at b = 0.881 (left) and b = 0.948 (right). Real parts are shown by solid lines and imaginary parts are shown by dashed ones. Amplitude profile of soliton marked by f 1 is shown in (f). To distinguish with Re(ϕf1), it is enhanced by 0.1.

Fig. 3
Fig. 3

Instability growth rate of even solitons in the semi-infinite gap. (b) Unstable propagation of soliton at b = 0.948 marked by f1 in Fig. 2(a). (c, d) Power-flow densities of solitons at b = 0.948 marked by c1 and d1 in Fig. 2(a). Inset: the corresponding stable and unstable propagations. Scaled real part of lattice is shown. The evolutions of amplitude profiles are plotted in (b–d).

Fig. 4
Fig. 4

(a) Power P of odd solitons versus b in the semi-infinite gap. Shaded region denotes the first band. (b–f) Profiles of solitons marked by dots in (a) at b = 0.884 (left) and b = 0.949 (right). Real parts are shown by solid lines and imaginary parts are shown by dashed ones. Amplitude profile of soliton marked by f1 is shown in (f). To distinguish with Re(ϕf1), it is enhanced by 0.1.

Fig. 5
Fig. 5

(a) Instability growth rate for odd solitons in the semi-infinite gap. (b) Power-flow density of soliton corresponding to (d). Scaled real part of lattice is shown. Stable (c) and unstable (d) propagations of solitons at b = 0.949 [marked by b1 and f1 in Fig. 4(a)]. The evolutions of amplitude profiles are plotted in (c, d).

Fig. 6
Fig. 6

Dependence of power P on b for even (a) and odd (b) solitons with a different number of humps in the first gap. Shaded regions denote the first and second bands. Solid curves correspond to the stable solitons.

Fig. 7
Fig. 7

Profiles of even solitons (a–c) and odd solitons (d–f) at b = 1.40 marked by dots in Fig. 6. Examples of amplitude profiles are shown in (c, f).

Fig. 8
Fig. 8

Stable propagations of multi-stable solitons marked by hollow circles in Fig. 6. (a–c) Even solitons. (d–f) Odd solitons. b = 2.13 in all panels. The evolutions of amplitude profiles are plotted.

Equations (3)

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i ψ z + 1 2 2 ψ x 2 [ V ( x ) + i W ( x ) ] ψ + | ψ | 2 ψ | ψ | 4 ψ = 0 .
1 2 d 2 ϕ d x 2 + b ϕ [ V ( x ) + i W ( x ) ] ϕ + | ϕ | 2 ϕ | ϕ | 4 ϕ = 0 ,
i ( i W i Im ( ϕ 2 ) ( 1 2 | ϕ | 2 ) L ^ Re ( ϕ 2 ) ( 1 2 | ϕ | 2 ) L ^ + Re ( ϕ 2 ) ( 1 2 | ϕ | 2 ) i W + i Im ( ϕ 2 ) ( 1 2 | ϕ | 2 ) ) ( v w ) = λ ( v w ) ,

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