Abstract

We report on the existence and stability of mixed-gap vector solitons in parity-time (PT)-symmetric mixed linear–nonlinear optical lattices. The first component is single-peaked, and the propagation constant is in the semi-infinite gap. The second component is the out-of-phase dipole mode; its propagation constant belongs to the first finite gap. The imaginary part and the depth of the PT-symmetric nonlinear optical lattice will significantly affect the existence and stability domains of these vector solitons. The propagation constant of the first component can also influence the existence and stability of the vector solitons. Finally, we also study the effect of the PT-symmetric linear optical lattice on the vector solitons’ stability.

© 2014 Optical Society of America

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2013 (9)

2012 (10)

Y. V. Izdebskaya, J. Rebling, A. S. Desyatnikov, and Y. S. Kivshar, “Observation of vector solitons with hidden vorticity,” Opt. Lett. 37, 767–769 (2012).
[CrossRef]

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20, 16823–16831 (2012).

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (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]

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

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. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85, 043826 (2012).
[CrossRef]

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

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

2011 (6)

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

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[CrossRef]

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

H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (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 superlattices,” Opt. Lett. 36, 2680–2682 (2011).
[CrossRef]

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

2010 (4)

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35, 2976–2978 (2010).
[CrossRef]

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010).
[CrossRef]

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

2009 (2)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[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]

2008 (2)

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

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

2007 (2)

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]

R. EI-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]

2006 (2)

2005 (1)

2004 (3)

Z. Chen, A. Bezryadina, I. Makasyuk, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. 29, 1656–1658 (2004).
[CrossRef]

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

G. D. Montesinos, V. M. Pérez-García, and H. Michinel, “Stabilized two-dimensional vector solitons,” Phys. Rev. Lett. 92, 133901 (2004).
[CrossRef]

2003 (2)

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef]

2002 (1)

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

2001 (1)

A. S. Desyatnikov and Y. S. Kivshar, “Necklace-ring vector solitons,” Phys. Rev. Lett. 87, 033901 (2001).
[CrossRef]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

1998 (2)

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

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
[CrossRef]

1988 (1)

Abdullaev, F. K.

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

F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010).
[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]

Andrews, M. R.

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
[CrossRef]

Bender, C. M.

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

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

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

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

Bezryadina, A.

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[CrossRef]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

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

Brody, D. C.

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

Chen, W.

Chen, X.

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]

Z. Chen, A. Bezryadina, I. Makasyuk, and J. Yang, “Observation of two-dimensional lattice vector solitons,” Opt. Lett. 29, 1656–1658 (2004).
[CrossRef]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

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

C. E. Rüter, K. G. Makris, R. EI-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]

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

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

R. EI-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]

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Vector solitons in birefringent nonlinear dispersive media,” Opt. Lett. 13, 53–55 (1988).
[CrossRef]

Cohen, O.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Denschlag, J. H.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
[CrossRef]

Desyatnikov, A. S.

Dmitriev, S. V.

Dong, L.

Driben, R.

Duchesne, D.

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

EI-Ganainy, R.

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

C. E. Rüter, K. G. Makris, R. EI-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. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

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

R. EI-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]

Fleischer, J. W.

O. Cohen, T. Schwartz, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Multiband vector lattice solitons,” Phys. Rev. Lett. 91, 113901 (2003).
[CrossRef]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[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]

Grimm, R.

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X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38, 2723–2725 (2013).
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X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30, 1987–1995 (2013).
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H. Wang, W. He, L. Zheng, X. Zhu, H. Li, and Y. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45, 245401 (2012).

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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).
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X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38, 2723–2725 (2013).
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X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30, 1987–1995 (2013).
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H. Wang, W. He, L. Zheng, X. Zhu, H. Li, and Y. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45, 245401 (2012).

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
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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 superlattices,” Opt. Lett. 36, 2680–2682 (2011).
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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).
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C. E. Rüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
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G. D. Montesinos, V. M. Pérez-García, and H. Michinel, “Stabilized two-dimensional vector solitons,” Phys. Rev. Lett. 92, 133901 (2004).
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B. Midya and R. Roychoudhury, “Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87, 045803 (2013).
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S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
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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).
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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).
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A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly PT-symmetric systems: spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
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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).
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K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
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Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
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R. EI-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
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S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
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A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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G. D. Montesinos, V. M. Pérez-García, and H. Michinel, “Stabilized two-dimensional vector solitons,” Phys. Rev. Lett. 92, 133901 (2004).
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A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
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B. Midya and R. Roychoudhury, “Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87, 045803 (2013).
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M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
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C. E. Rüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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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).
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F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010).
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C. E. Rüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
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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).
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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).
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S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
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S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
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Sukhorukov, A. A.

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M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
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M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
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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).
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Wang, H.

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M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. H. Denschlag, “Tuning the scattering length with an optically induced Feshbach resonance,” Phys. Rev. Lett. 93, 123001 (2004).
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Yang, J.

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

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F. K. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E 82, 056606 (2010).
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D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
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F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
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H. Wang, W. He, L. Zheng, X. Zhu, H. Li, and Y. He, “Defect gap solitons in real linear periodic optical lattices with parity-time-symmetric nonlinear potentials,” J. Phys. B 45, 245401 (2012).

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 superlattices,” Opt. Lett. 36, 2680–2682 (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. A 85, 043826 (2012).
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Zhu, X.

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30, 1987–1995 (2013).
[CrossRef]

X. Zhu, H. Wang, H. Li, W. He, and Y. He, “Two-dimensional multipeak gap solitons supported by parity-time-symmetric periodic potentials,” Opt. Lett. 38, 2723–2725 (2013).
[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. A 85, 013831 (2012).
[CrossRef]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

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

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]

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 superlattices,” Opt. Lett. 36, 2680–2682 (2011).
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J. Math. Phys. (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

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

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
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Nature (2)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kun, and W. Ketterle, “Observation of Feshbach resonances in a Bose-Einstein condensate,” Nature 392, 151–154 (1998).
[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).
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Opt. Express (4)

Opt. Lett. (12)

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Figures (11)

Fig. 1.
Fig. 1.

When V1=6cos(2x) and W1=2.1sin(2x). (a) PT-symmetric linear periodic potential. (b) Corresponding band structure.

Fig. 2.
Fig. 2.

For V1=6cos(2x), W1=2.1sin(2x), V2=cos2(x), W2=sin(2x), and μ1=5.0. (a) Total power of vector solitons versus propagation constant μ2 (shaded regions are Bloch bands). (b) Power diagrams of fundamental (P1) and out-of-phase dipole (P2) components.

Fig. 3.
Fig. 3.

When V1=6cos(2x), W1=2.1sin(2x), V2=cos2(x), W2=sin(2x), and μ1=5.0. (a) max[Re(δ)] versus μ2. (b), (c) Profiles of the first component (solid line is the real part, while the dashed line is the imaginary part) and the second component of the vector soliton when μ2=0.9. (d), (e) Corresponding stable propagations of the two perturbed components. (f), (g) Profiles of the two components of the vector soliton when μ2=0. (h), (i) Corresponding unstable propagations.

Fig. 4.
Fig. 4.

When V1=6cos(2x), W1=2.1sin(2x), V2=cos2(x), W2=2.35sin(2x), and μ1=5.0. (a) Power of vector solitons. (b) Power of first component (P1) and second component (P2).

Fig. 5.
Fig. 5.

For V1=6cos(2x), W1=2.1sin(2x), V2=cos2(x), W2=2.35sin(2x), and μ1=5.0. (a) Diagram of max[Re(δ)]. (b), (c) Profiles of the two components and (d), (e) stable propagations of the two perturbed components for μ2=0.9. (f), (g) Profiles of the two components when μ2=1.0. (h), (i) Unstable propagations of the two perturbed components.

Fig. 6.
Fig. 6.

When V1=6cos(2x). (a). Vector soliton profile for W1=2.1sin(2x), V2=3cos2(x), W2=3sin(2x), μ1=5.0, and μ2=0. (b), (c) Corresponding stable propagations of the two perturbed components. (d). Profile of the vector soliton for W1=2.1sin(2x), V2=cos2(x), W2=sin(2x), μ1=3.5, and μ2=1.9. (e), (f) Stable propagations of the two perturbed components. (g) Vector soliton profile when W1=3.6sin(2x), V2=cos2(x), W2=sin(2x), μ1=5.0, and μ2=0.9. (h), (i) Unstable propagations of the two perturbed components.

Fig. 7.
Fig. 7.

When V1=6cos(2x), W1=5.4sin(2x), V2=cos2(x), W2=sin(2x), μ1=5.0, and μ2=0.9. (a), (b) Shapes of two components of the vector soliton. (c), (d) Corresponding unstable propagations of the two perturbed components.

Fig. 8.
Fig. 8.

For V1=6cos(2x), W1=5.88sin(2x), V2=cos2(x), W2=sin(2x), μ1=5.0, and μ2=0.9. The profiles of the two components of the vector soliton are depicted in (a) and (b), respectively. The unstable propagations of the two perturbed components are displayed in (c) and (d), respectively.

Fig. 9.
Fig. 9.

WhenV1=0, W1=0, μ1=5.0, and μ2=0.9. The profile of the vector soliton and the stable propagations of the two perturbed components when V2=cos2(x) and W2=0.1sin(2x) are shown in (a), (b), and (c), respectively. (d), (e), and (f) exhibit the vector soliton profile and the unstable propagations of the two perturbed components for V2=cos2(x) and W2=0.2sin(2x), respectively. (g), (h), and (i) show the profile of the vector soliton and the unstable propagations of the two perturbed components when V2=3cos2(x) and W2=0.3sin(2x), respectively.

Fig. 10.
Fig. 10.

When V1=0, W1=0, μ1=6.5, and μ2=1.2. (a), (b) Two components of the vector soliton for V2=cos2(x) and W2=0.1sin(2x). (c), (d) Corresponding stable propagations of the two perturbed components.

Fig. 11.
Fig. 11.

In defocusing media. For V1=6cos(2x), W1=2.1sin(2x), and V2=cos2(x). (a) and (b) depict the profile of the first component of the vector soliton and the stability of the perturbed component for W2=sin(2x) and μ1=μ2=0.3. The profile of the first component and the unstable propagation of the perturbed component when W2=3sin(2x) and μ1=μ2=0.3 are shown in (c) and (d), respectively.

Equations (6)

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iU1,2z+2U1,2x2+[V1(x)+iW1(x)]U1,2+[1+V2(x)+iW2(x)]U1,2(|U1|2+|U2|2)=0.
2q1,2x2+(V1+iW1)q1,2+(1+V2+iW2)q1,2(|q1|2+|q2|2)μ1,2q1,2=0.
U1,2(x,z)=exp(iμ1,2z)[q1,2+F1,2exp(δz)+G1,2*exp(δ*z)].
{F1δ=i{L^1F1+(1+V2+iW2)[(2|q1|2+|q2|2)F1+q12G1+q1q2*F2+q1q2G2]},G1δ=i{L^1*G1(1+V2iW2)[(q12)*F1+(2|q1|2+|q2|2)G1+q1*q2*F2+q1*q2G2]},F2δ=i{L^2F2+(1+V2+iW2)[q1*q2F1+q1q2G1+(|q1|2+2|q2|2)F2+q22G2]},G2δ=i{L^2*G2(1+V2iW2)[q1*q2*F1+q1q2*G1+(q22)*F2+(|q1|2+2|q2|2)G2]}.
iU1,2z+2U1,2x2+[V1(x)+iW1(x)]U1,2[1+V2(x)iW(x)]U1,2(|U1|2+|U2|2)=0.
2q1,2x2+[V1(x)+iW1(x)]q1,2[1+V2(x)iW2(x)]q1,2(|q1|2+|q2|2)μ1,2q1,2=0.

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