Abstract

We report the evolution of higher-order nonlinear states in a focusing cubic medium, where both the linear refractive index and the nonlinearity are spatially modulated by a complex optical lattice exhibiting a parity-time (𝒫𝒯) symmetry. We reveal that introduction of out-of-phase nonlinearity modulation makes possible the stabilization of higher-order solitons with number of poles up to 7, which are highly unstable in linear 𝒫𝒯 lattices. Under appropriate conditions, multipole-mode solitons with out-of-phase components in the neighboring lattice sites are completely stable provided that their power or propagation constant exceeds a critical value. Thus, our findings suggest an effective way for the realization of stable multipole-mode solitons in periodic potentials with gain-loss components.

© 2013 OSA

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    [CrossRef]
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  4. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett.33, 1747–1749 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 013831 (2012).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  29. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).
  30. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math.118, 153–197 (2007).
    [CrossRef]
  31. N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).
  32. M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
    [CrossRef]
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2013 (1)

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯 -symmetric periodic potentials,” Phys. Rev. A87, 013812 (2013).
[CrossRef]

2012 (8)

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A13, 46–54 (2012).

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys.64, 1243–1258 (2012).

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 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. A85, 043826 (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]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 013831 (2012).
[CrossRef]

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express20, 16823–16831 (2012).

2011 (7)

H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express19, 4030–4035 (2011).
[CrossRef] [PubMed]

Z. Lu and Z.-M. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express19, 11457–11462 (2011).
[CrossRef] [PubMed]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A84, 053855 (2011).
[CrossRef]

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

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 041805 (2011).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

2010 (2)

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]

2009 (3)

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, 030402 (2008).
[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, 1747–1749 (2008).
[CrossRef] [PubMed]

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (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, 153–197 (2007).
[CrossRef]

2005 (1)

2004 (2)

2003 (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[CrossRef]

2001 (1)

B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001).
[CrossRef]

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 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]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

Bagchi, B.

B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001).
[CrossRef]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

Chen, D.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A13, 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. A13, 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. A85, 013831 (2012).
[CrossRef]

Christodoulides, D. N.

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]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

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

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[CrossRef]

Dong, L.

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express20, 16823–16831 (2012).

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011).
[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]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[CrossRef]

El-Ganainy, R.

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]

Ellis, F. M.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[CrossRef]

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 023822 (2012).
[CrossRef]

Gocalek, J.

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[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 and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯 -symmetric periodic potentials,” Phys. Rev. A87, 013812 (2013).
[CrossRef]

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys.64, 1243–1258 (2012).

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 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, 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. A85, 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. A85, 043826 (2012).
[CrossRef]

Huang, C.

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011).
[CrossRef]

Infeld, E.

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[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. A84, 053855 (2011).
[CrossRef]

Kartashov, Y. V.

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]

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).

Konotop, V. V.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 041805 (2011).
[CrossRef]

Kottos, T.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

Lakoba, T. I.

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

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

Lee, J. M.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

Li, A.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

Li, C.

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express20, 16823–16831 (2012).

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (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. A84, 053855 (2011).
[CrossRef]

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (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. A13, 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. A13, 46–54 (2012).

Lin, Z.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

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. A85, 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.

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. A85, 043826 (2012).
[CrossRef]

Lu, Z.

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. A85, 043826 (2012).
[CrossRef]

Makris, K. G.

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]

Malomed, B. A.

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

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Two-dimensional solitons in nonlinear lattices,” Opt. Lett.34, 770–772 (2009).
[CrossRef] [PubMed]

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

Matuszewski, M.

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

Mihalache, D.

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯 -symmetric periodic potentials,” Phys. Rev. A87, 013812 (2013).
[CrossRef]

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys.64, 1243–1258 (2012).

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 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]

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. H.

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]

Nixon, S.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 023822 (2012).
[CrossRef]

Oberthaler, M.

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

Quesne, C.

B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001).
[CrossRef]

Ramezani, H.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

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]

Schindler, J.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

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]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[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. A84, 053855 (2011).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[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]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

Torner, L.

Trippenbach, M.

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

Vakhitov, N. G.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).

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]

Vysloukh, V. A.

Wang, H.

Wang, J.

Xu, Z.

Yang, J.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 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, 153–197 (2007).
[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. A13, 46–54 (2012).

Ye, F.

Zezyulin, D. A.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 041805 (2011).
[CrossRef]

Zhang, Z.-M.

Zheng, M. C.

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

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. A85, 043826 (2012).
[CrossRef]

Zhong, S.

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011).
[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. A85, 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. A84, 053855 (2011).
[CrossRef]

Znojil, M.

B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz.16, 783–789 (1973).

J. Phys. A (1)

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys. A45, 444029 (2012).
[CrossRef]

Mod. Phys. Lett. (1)

B. Bagchi, C. Quesne, and M. Znojil, “Generalized continuity equation and modified normalization in PT-dymmetric quantum mechanics,” Mod. Phys. Lett.16, 2047–2057 (2001).
[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]

Nature (London) (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature (London)422, 147–150 (2003).
[CrossRef]

Opt. Commun. (1)

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]

Opt. Express (5)

Opt. Lett. (5)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463, 1 – 126 (2008).
[CrossRef]

Phys. Rev. A (9)

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A84, 043830 (2011).
[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. A85, 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. A84, 053855 (2011).
[CrossRef]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A85, 013831 (2012).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A83, 041805 (2011).
[CrossRef]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A85, 023822 (2012).
[CrossRef]

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯 -symmetric periodic potentials,” Phys. Rev. A87, 013812 (2013).
[CrossRef]

J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with 𝒫𝒯 symmetries,” Phys. Rev. A84, 040101 (2011).
[CrossRef]

M. Trippenbach, E. Infeld, J. Gocalek, M. Matuszewski, M. Oberthaler, and B. A. Malomed, “Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps,” Phys. Rev. A78, 013603 (2008).
[CrossRef]

Phys. Rev. Lett. (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]

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]

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. A13, 46–54 (2012).

Rev. Mod. Phys. (1)

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

Rom. Rep. Phys. (1)

Y. He and D. Mihalache, “Spatial solitons in parity-time-symmetric mixed linear-nonlinear optical lattices: Recent theoretical results,” Rom. Rep. Phys.64, 1243–1258 (2012).

Stud. Appl. Math. (1)

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

Other (1)

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” ArXiv e-prints (2013).

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Figures (6)

Fig. 1
Fig. 1

(a) Band-gap structure of lattice. Solid: χ = 0.2; dashed: χ = 0.5. (b) Power P and amplitude ratio of fundamental solitons versus propagation constant b. (c, d) Soliton profiles at b = 2.3 and 9.0, real parts are plotted in blue while imaginary parts are plotted in red. (e) Transverse power flow across the lattice. (f) Propagation of soliton at b = 9.0. p = 4, Ω = 4, σ = 0.2 in all panels and χ = 0.2 except for (a).

Fig. 2
Fig. 2

(a) Power P (solid) and amplitude ratio (dashed) of dipole solitons versus b. (b, c) Soliton profiles at b = 2.74 and 5.70. (d) Instability growth rate versus b. (e, f) Unstable and stable propagations of dipole solitons marked in (d) at b = 3.44 and 5.70, respectively. σ = 0.2, χ = 0.2 in all panels.

Fig. 3
Fig. 3

(a) Power P (solid) and amplitude ratio (dashed) of triple solitons versus b. (b, c) Soliton profiles at b = 2.80 and 3.66. (d) Instability growth rate versus b. (e, f) Unstable and stable propagations of dipole solitons marked in (d) at b = 3.28 and 3.66, respectively. σ = 0.2, χ = 0.2 in all panels.

Fig. 4
Fig. 4

(a) Profiles of dipole solitons at b = 5.0 in lattices with σ = 0.2. Solid: χ = 0.2; Dashed: χ = 0.4. (b) Profiles of triple solitons at b = 4.8 in lattices with χ = 0.2. Solid: σ = 0.2; Dashed: σ = 0.4. (c) Pb diagram of dipole solitons in lattices with χ = 0.4, σ = 0.2. (d) Pb diagram of triple solitons in lattices with χ = 0.2, σ = 0.4. The stable branches are plotted by solid curves and the unstable branches are plotted by dashed curves.

Fig. 5
Fig. 5

Profiles and stable propagations of solitons with 5– (a) and 7– (b) poles at b = 5.4, σ = 0.1. Symmetric (c) and antisymmetric (d) dual-core solitons at b = 2.3, σ = 0.2 and their stable propagations.

Fig. 6
Fig. 6

Excitation of fundamental (a) and dipole (b) solitons by Gaussian beams with A = 1.7, σ = 0.1, θ = 0.1. d = 1 in (a) and 3 in (b). The lattices are removed at z = 50 and 60, respectively. (c, d) Asymmetrical unpacking of 3– and 5–pole solitons derived numerically at b = 4.3 and 5.4, respectively. χ = 0.2, σ = 0.1 in all panels.

Equations (3)

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

i q z = 1 2 2 q x 2 | q | 2 q + σ R ( x ) | q | 2 q p R ( x ) q ,
1 2 2 u x 2 b u + | u | 2 u σ R | u | 2 u + p R u = 0 ,
i λ v = ( i [ σ W Re ( u 2 ) + ( σ V 1 ) Im ( u 2 ) p W + 2 σ W | u | 2 ] ) v + ( 1 2 2 x 2 + 2 | u | 2 b + p V 2 σ V | u | 2 + ( σ V 1 ) Re ( u 2 ) σ W Im ( u 2 ) ) w i λ w = ( i [ σ W Re ( u 2 ) ( σ V 1 ) Im ( u 2 ) p W + 2 σ W | u | 2 ] ) w + ( 1 2 2 x 2 + 2 | u | 2 b + p V 2 σ V | u | 2 ( σ V 1 ) Re ( u 2 ) + σ W Im ( u 2 ) ) v .

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