Abstract

We study the families of solitons supported by parity time (PT) symmetric potentials with spatially modulated nonlinearity. The competition between the real part of PT symmetric potentials and out-of-phase nonlinearity modulations can result in remarkable power-dependent shape transformations of solitons and substantially modifies their stability properties. Linear stability analysis reveals that, with the increase of the strength of out-of-phase nonlinearity modulation, the odd soliton’s stability domain deceases, while even for solitons, the reverse applies. The effect of variation of the amplitude of the imaginary part of PT symmetric potential on stability properties of solitons is also studied.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  32. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Stability of solitons in PT-symmetric nonlinear potentials,” Europhys. Lett. 96, 64003 (2011).
    [CrossRef]
  33. 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]
  34. 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]
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    [CrossRef]
  36. M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear system,” Opt. Lett. 30, 2140–2142 (2005).
    [CrossRef]
  37. 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]
  38. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
    [CrossRef]

2012 (3)

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]

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

2011 (11)

F. Kh. 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,” Europhys. Lett. 96, 64003 (2011).
[CrossRef]

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A 84, 043830 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

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

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

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

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

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[CrossRef]

H. C. Huang, Y. J. He, Y. Z. Liu, and H. Z. Wang, “Dynamics and all-optical control of solitons at the interface of optical superlattices with spatially modulated nonlinearity,” Opt. Express 19, 8795–8801 (2011).
[CrossRef]

2010 (4)

2009 (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]

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]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Stabilization of multibeam necklace solitons in circular arrays with spatially modulated nonlinearity,” Phys. Rev. A 80, 053816 (2009).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Vector solitons in nonlinear lattices,” Opt. Lett. 34, 3625–3627 (2009).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328–11334 (2009).
[CrossRef]

L. Torner and Y. V. Kartashov, “Light bullets in optical tandems,” Opt. Lett. 34, 1129–1131 (2009).
[CrossRef]

2008 (5)

2007 (3)

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]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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

2006 (1)

2005 (2)

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

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

2004 (1)

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]

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, 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, 093902 (2009).
[CrossRef]

Belmonte-Beitia, J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Bender, C. M.

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

Blömer, D.

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]

Chen, Z.

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

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]

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

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

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]

E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice soliton,” Opt. Lett. 29, 845–847 (2004).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

Dong, L.

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A 84, 043830 (2011).
[CrossRef]

L. Dong and H. Li, “Surface solitons in nonlinear lattices,” J. Opt. Soc. Am. B 27, 1179–1183 (2010).
[CrossRef]

Dreisow, F.

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]

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]

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

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

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]

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 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]

Guo, Z.

He, Y.

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]

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

He, Y. J.

Heinrich, M.

Hu, B.

Hu, S.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[CrossRef]

Hu, W.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[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. A 84, 043830 (2011).
[CrossRef]

Huang, H. C.

Jiang, X.

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

Kartashov, Y. V.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

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

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

L. Torner and Y. V. Kartashov, “Light bullets in optical tandems,” Opt. Lett. 34, 1129–1131 (2009).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328–11334 (2009).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Vector solitons in nonlinear lattices,” Opt. Lett. 34, 3625–3627 (2009).
[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]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Stabilization of multibeam necklace solitons in circular arrays with spatially modulated nonlinearity,” Phys. Rev. A 80, 053816 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index,” Opt. Lett. 33, 2173–2175 (2008).
[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, 1747–1749 (2008).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, T. Pertsch, and L. Torner, “Surface solitons at interfaces of arrays with spatially modulated nonlinearity,” Opt. Lett. 33, 1120–1122 (2008).
[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]

Konotop, V. V.

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

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

Lakoba, T. I.

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

Li, C.

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A 84, 043830 (2011).
[CrossRef]

Li, H.

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, 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]

Liu, S.

Liu, Y. Z.

Lu, D.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[CrossRef]

Lu, Z.

Ma, X.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[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 lattices,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

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

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]

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Vector solitons in nonlinear lattices,” Opt. Lett. 34, 3625–3627 (2009).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Stabilization of multibeam necklace solitons in circular arrays with spatially modulated nonlinearity,” Phys. Rev. A 80, 053816 (2009).
[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]

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]

Y. J. 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, 1716–1718 (2010).
[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]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

Musslimani, Z. H.

Nixon, S.

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

Nolte, S.

Pérez-García, V. M.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Pertsch, T.

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[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]

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]

Shankar, R.

R. Shankar, Principles of Quantum Mechanics (Springer, 1994).

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]

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

Stegeman, G. I.

Szameit, A.

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328–11334 (2009).
[CrossRef]

L. Torner and Y. V. Kartashov, “Light bullets in optical tandems,” Opt. Lett. 34, 1129–1131 (2009).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Vector solitons in nonlinear lattices,” Opt. Lett. 34, 3625–3627 (2009).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Stabilization of multibeam necklace solitons in circular arrays with spatially modulated nonlinearity,” Phys. Rev. A 80, 053816 (2009).
[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]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index,” Opt. Lett. 33, 2173–2175 (2008).
[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, 1747–1749 (2008).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, T. Pertsch, and L. Torner, “Surface solitons at interfaces of arrays with spatially modulated nonlinearity,” Opt. Lett. 33, 1120–1122 (2008).
[CrossRef]

Torres, P. J.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

Tünnermann, A.

Ultanir, E. A.

Vekslerchik, V.

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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]

Vysloukh, V. A.

Wang, H.

Wang, H. Z.

Wang, J.

Yang, J.

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

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

Yang, Z.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[CrossRef]

Ye, F.

Zezyulin, D. A.

F. Kh. 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,” Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Zhang, Z.

Zheng, L.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[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. A 84, 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. 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]

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

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

Europhys. Lett. (1)

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

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

J. Phys. A (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[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. (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. (13)

Y. J. 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, 1716–1718 (2010).
[CrossRef]

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

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

E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice soliton,” Opt. Lett. 29, 845–847 (2004).
[CrossRef]

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]

L. Torner and Y. V. Kartashov, “Light bullets in optical tandems,” Opt. Lett. 34, 1129–1131 (2009).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, T. Pertsch, and L. Torner, “Surface solitons at interfaces of arrays with spatially modulated nonlinearity,” Opt. Lett. 33, 1120–1122 (2008).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Vector solitons in nonlinear lattices,” Opt. Lett. 34, 3625–3627 (2009).
[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]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index,” Opt. Lett. 33, 2173–2175 (2008).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear system,” Opt. Lett. 30, 2140–2142 (2005).
[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, 1747–1749 (2008).
[CrossRef]

Phys. Rev. A (7)

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 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, 041805 (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. A 85, 013831 (2012).
[CrossRef]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Stabilization of multibeam necklace solitons in circular arrays with spatially modulated nonlinearity,” Phys. Rev. A 80, 053816 (2009).
[CrossRef]

L. Dong, H. Li, C. Huang, S. Zhong, and C. Li, “Higher-charged vortices in mixed linear-nonlinear circular arrays,” Phys. Rev. A 84, 043830 (2011).
[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]

S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
[CrossRef]

Phys. Rev. Lett. (5)

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef]

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

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

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

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011).
[CrossRef]

Stud. Appl. Math. (1)

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

Other (1)

R. Shankar, Principles of Quantum Mechanics (Springer, 1994).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Typical PT potential prV(x)+ipiW(x) and associated nonlinearity modulation 1σV(x). Solid blue, real part; dotted red, imaginary part; dashed green, nonlinearity modulation. (b) Bandgap structure of the PT potential corresponding to (a). In all panels, pr=4, pi=0.4, and σ=0.5.

Fig. 2.
Fig. 2.

(a)–(c) Complex fields of odd solitons with b=2.4, 4.6, and 10 at σ=0.5 [marked by dots in Fig. 3(a)], respectively (solid blue, real part; dashed red, imaginary part). (d)–(f) The complex fields of odd solitons with b=2.4, 4.6, and 10 at σ=0.85 [marked by dots in Fig. 3(a)], respectively. In shaded regions V1/2, while in the white regions V<1/2. In all panels, pr=4 and pi=0.4.

Fig. 3.
Fig. 3.

(a) U versus b for odd solitons at σ=0.5 (lower) and σ=0.85 (upper). Dashed curves denote the unstable solitons. (b) Stability (white) and instability (shaded) area on the (σ,b) plane for odd solitons. (c) δr versus b for odd solitons at σ=0.5 (lower) and σ=0.85 (upper). (d) The modulus u(x) and the current density j(x) for odd solitons corresponding to Fig. 2(f). In all panels, pr=4 and pi=0.4.

Fig. 4.
Fig. 4.

U versus pi for odd solitons with (a) b=4.6 and (b) b=10. (c) Stability (white) and instability (shaded) area on the (pi,b) plane for odd solitons with pr=4 at σ=0.5. The complex fields of odd solitons with (d) b=4.6 and (e) b=10 for different pi [marked by dots in Figs. 3(a) and 3(b)], respectively. Real parts are shown by solid curves and imaginary parts are shown by dashed ones. In all panels except (c), pr=4 and σ=0.85.

Fig. 5.
Fig. 5.

Complex fields of even solitons with (a) b=2.8 and (b) b=6 for different σ [marked by dots in Fig. 6(a)], respectively. Real parts are shown by solid curves and imaginary parts are shown by dashed ones. In all panels, pr=4 and pi=0.4.

Fig. 6.
Fig. 6.

(a) U versus b for even solitons at σ=0.3 (lower) and σ=0.6 (upper). Dashed curves denote the unstable solitons. (b) Stability (white) and instability (shaded) area on the (σ,b) plane for even solitons. bo is the upper edge of the stability domain of odd solitons. (c) δr versus b for even solitons at σ=0.3 (lower) and σ=0.6 (upper). (d) The modulus u(x) and the current density j(x) for even solitons with b=2.8 at σ=0.6, which corresponds to Fig. 5(a). In all panels, pr=4 and pi=0.4.

Fig. 7.
Fig. 7.

U versus pi for even solitons with (a) b=2.8 and (b) b=6. (c) Dependence of blow on pi for even solitons with pr=4 at σ=0.6, where blow is the lower edge of the stability domain of even solitons. (d) The complex fields of even solitons with b=2.8 for different pi [marked by dots in Fig. 7(a)], respectively. Real parts are shown by solid curves and imaginary parts are shown by dashed ones. In all panels except (c), pr=4 and σ=0.3.

Equations (9)

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

iqz+2qx2+[1σV(x)]|q|2q+[prV(x)+ipiW(x)]q=0.
V(x)=cos2(2x)andW(x)=sin(4x).
bw=d2wdx2+(1σV)|w|2w+(prV+ipiW)w.
lim|x||w(x)|=0.
bu=d2udx2j2u3+(1σV)u3+prVu,
djdx=piWu2,
bU=|wx|2dx+(1σV)|w|4dx+prV|w|2dx,
q(x,z)={w(x)+[g(x)t(x)]exp(δz)+[g(x)+t(x)]*exp(δ*z)}exp(ibz),
i(piWiγIm(w2)L0γRe(w2)L0+γRe(w2)piW+iγIm(w2))(gt)=δ(gt),

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