Abstract

We report the stability of in-phase quadruple and off-site vortex solitons in the parity-time-symmetric periodic potentials with defocusing Kerr nonlinearity. All solitons can exist in the first gap and can be stable in a certain range. It is shown that the power of vortex solitons decreases and the stable region shrinks with increase of the topological charge. Especially the stable region is very small for double charge vortex solitons. The power evolutions of vortex solitons along the propagation distance are also analysed. Increasing the lattice depth or decreasing the gain-loss component can stabilize vortex solitons. For both lattice depth and gain-loss component there exists a critical value, below or above which all vortex solitons will become unstable.

© 2014 Optical Society of America

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References

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  7. D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
    [Crossref] [PubMed]
  8. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
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  9. J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
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  10. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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  11. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear Lattices,” Phys. Rev. A 83(4), 041805 (2011).
    [Crossref]
  12. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
    [Crossref]
  13. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
    [Crossref]
  14. H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
    [Crossref]
  15. X. Zhu, H. Wang, H. Li, W. He, and Y. J. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38(15), 2723–2725 (2013).
    [Crossref] [PubMed]
  16. H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
    [Crossref]
  17. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  23. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [Crossref] [PubMed]
  24. J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118(2), 153 (2007).
    [Crossref]
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2014 (3)

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

2013 (2)

2012 (4)

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

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

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

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

2011 (2)

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

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

2008 (2)

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

2007 (1)

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

2004 (3)

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

2003 (1)

2001 (1)

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001).
[Crossref] [PubMed]

1998 (1)

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

Abdullaev, F. K.

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

Achilleos, V.

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

Alexander, T. J.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Bartal, G.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Bender, C. M.

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

Boettcher, S.

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

Carretero-González, R.

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

Chen, Z.

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

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 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(3), 030402 (2008).
[Crossref] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Cohen, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Dong, L.

El-Ganainy, R.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

Fleischer, J. W.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Frantzeskakis, D. J.

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

Ge, L.

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

He, W.

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

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

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

He, Y.

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

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

He, Y. J.

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

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

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

Huang, C.

Hudock, J.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

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(5), 053855 (2011).
[Crossref]

Kartashov, Y. V.

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

Kevrekidis, P. G.

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

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001).
[Crossref] [PubMed]

Kivshar, Y. S.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Konotop, V. V.

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

Lai, T. S.

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

Lakoba, T. I.

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

Lee, C. H.

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

Li, C.

Li, H.

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

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

Li, H. G.

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

Ling, D.

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

Liu, J.

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

Makasyuk, I.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 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(3), 030402 (2008).
[Crossref] [PubMed]

Malomed, B. A.

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001).
[Crossref] [PubMed]

Manela, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Martin, H.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Mihalache, D.

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

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 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(3), 030402 (2008).
[Crossref] [PubMed]

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003).
[Crossref] [PubMed]

Neshev, D. N.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Nixon, S.

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

Ostrovskaya, E. A.

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

Segev, M.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Shi, S.

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[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(5), 053855 (2011).
[Crossref]

Shi, Z. W.

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

Wang, H.

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

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

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

Yang, J.

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

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

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003).
[Crossref] [PubMed]

Zezyulin, D. A.

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

Zhang, S.

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

Zheng, L. X.

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

Zhu, X.

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

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

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

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

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

Chin. Phys. B (1)

H. Wang, D. Ling, S. Zhang, X. Zhu, and Y. He, “Gap solitons in parity-time complex superlattice with dual periods,” Chin. Phys. B 23(6), 064208 (2014).
[Crossref]

J. Phys. B (1)

H. Wang, W. He, L. X. Zheng, X. Zhu, H. G. Li, and Y. J. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45(24), 245401 (2012).
[Crossref]

New J. Phys. (1)

J. Yang, “Stability of vortex solitons in a photorefractive optical lattice,” New J. Phys. 6, 47 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (6)

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

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

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

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

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

H. G. Li, X. Zhu, Z. W. Shi, B. A. Malomed, T. S. Lai, and C. H. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89(5), 053811 (2014).
[Crossref]

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

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(2), 026601 (2001).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
[Crossref] [PubMed]

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

D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004).
[Crossref] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, and D. N. Christodoulides, “Observation of vortex-ring “discrete” solitons in 2D photonic lattices,” Phys. Rev. Lett. 92(12), 123904 (2004).
[Crossref] [PubMed]

Phys. Scr. (1)

H. Wang, W. He, S. Shi, X. Zhu, and Y. J. He, “Defect gap solitons in self-focusing Kerr media with parity-time-symmetric linear superlattices and modulated nonlinear lattices,” Phys. Scr. 89(2), 025502 (2014).
[Crossref]

Stud. Appl. Math. (1)

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

Other (5)

J. Yang, Nonlinear Waves in Integrable Systems, (SLAM, 2010).

L. M. Pismen, Vortices in Nonlinear Fields (Clarendon, 1999).

See M. S. Soskin and M. V. Vasnetsov, Progress in Optics, E. Wolf, ed. (Elsevier, 2001), Vol. 42.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Experimental results on vortex solitons in 2D photonic lattices were presented by M. Segev at CLEO Europe, Munich, Germany (2003), EE3-1.

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Figures (11)

Fig. 1
Fig. 1 The band structure of the PT-symmetric optical lattices associate with (a) W 0 =0.1 , (b) W 0 =0.5 , for all cases V 0 =9 .
Fig. 2
Fig. 2 Power P of in-phase quadruple solitons (red line) and vortex solitons with unit (blue line) and double charge (green line) versus propagation constant μ for V 0 =9, W 0 =0.1 . The stable branches are plotted by solid curves and the unstable branches are plotted by dashed curves. The black shaded region is the first Bloch bands. The inset shows power curves near the Bloch band.
Fig. 3
Fig. 3 (a) The profile of the real part of the in-phase quadruple soliton. (b) and (d) are the profiles ( | u | ) of the perturbed in-phase quadruple soliton for μ=11.96 at z=0 and z=500 , respectively. (c) The corresponding linear-stability spectrum.
Fig. 4
Fig. 4 The profile of in-phase quadruple soliton for μ=10.90 , (a) z=0 (b) z=300 . (c) Linear-stability spectrum.
Fig. 5
Fig. 5 (a) The profile of real part of unit charge vortex soliton, (b) the corresponding phase structure.
Fig. 6
Fig. 6 The evolution of unit charge vortex solitons, the first row is for a stable case ( μ=12.28 ). The second row is for an unstable case ( μ=11.50 ). The last row is the corresponding linear-stability spectrum.
Fig. 7
Fig. 7 The evolution of unit charge vortex solitons’ power versus the propagation distance, (a) is the stable case μ=12.28 . (b) is for the unstable case μ=11.50 .
Fig. 8
Fig. 8 The evolution of unit charge vortex soliton with V 0 =6, W 0 =0,μ=7.40 , (a) and (b) are evolution of unit charge vortex soliton at z=0 and z=130 ,respectively. (c) is its power evolution along the propagation distance and (d) is the corresponding linear-stability spectrum.
Fig. 9
Fig. 9 The stability domain of unit charge vortex solitons on W 0 values. The blue shade region is the stability domain of unit charge vortex solitons. Fixed V 0 =9 .
Fig. 10
Fig. 10 The phase of unit charge vortex soliton, the first row is for W 0 =0.1 and the second for W 0 =0.2 . The first column is for V 0 =9 and the second for V 0 =7 .
Fig. 11
Fig. 11 The real, imaginary part and phase of unit charge vortex solitons, the first row is for V 0 =9, W 0 =0.1 and the second for V 0 =9, W 0 =0.4 .

Equations (5)

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i U z + U xx + U yy +V(x,y)U | U | 2 U=0
V(x,y)= V 0 { [ sin 2 (x)+ sin 2 (y)]+i W 0 [sin(2x)+sin(2y)] },
u xx + u yy +V(x,y)u | u | 2 uμu=0
U(x,y,z)=exp(iμz){ u(x,y)+[p(x,y)q(x,y)]exp(λz)+ [p(x,y)+q(x,y)] * exp( λ * z) }
{ λp=i[i V 0 W 0 p+ 1 2 u 2 p 1 2 ( u * ) 2 pμq+ q xx + q yy + V 0 q2 | u | 2 q+ 1 2 u 2 q+ 1 2 ( u * ) 2 q] λq=i[μp+ p xx + p yy + V 0 p2 | u | 2 p 1 2 u 2 p 1 2 ( u * ) 2 pi V 0 W 0 q 1 2 u 2 q+ 1 2 ( u * ) 2 q]

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