Abstract

The existence and stability of vector solitons in non-parity-time (PT)-symmetric complex potentials are investigated. We study the vector soliton family, in which the propagation constants of the two components are different. It is found that vector solitons can be stable below and above the phase transition of the non-PT-symmetric complex potentials. Below the phase transition, vector solitons are stable in the low power region. Above the phase transition, there are two continuous stable intervals in the existence region. The profiles of two components of these vector solitons show the asymmetry and we also study the transverse power flow in the two components of these vector solitons in the non-PT-symmetric complex potentials.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2018 (1)

2017 (2)

J. Yang, “Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions,” Opt. Lett. 42(20), 4067–4070 (2017).
[Crossref] [PubMed]

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017).
[Crossref]

2016 (5)

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A (Coll. Park) 93(3), 031802 (2016).
[Crossref]

F. C. Moreira and S. B. Cavalcanti, “Localized modes in χ(2) media with non-PT-symmetric complex localized potentials,” Phys. Rev. A (Coll. Park) 94(4), 043818 (2016).
[Crossref]

S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136(4), 459–483 (2016).
[Crossref]

J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380(45), 3803–3809 (2016).
[Crossref]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

2015 (1)

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

2014 (3)

2013 (3)

2012 (8)

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (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]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref] [PubMed]

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(1), 013831 (2012).
[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]

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (2012).
[Crossref]

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20(15), 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(7410), 167–171 (2012).
[Crossref] [PubMed]

2011 (4)

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
[Crossref] [PubMed]

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]

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
[Crossref] [PubMed]

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

2010 (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

2009 (1)

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(9), 093902 (2009).
[Crossref] [PubMed]

2008 (1)

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]

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(17), 2632–2634 (2007).
[Crossref] [PubMed]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[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), 153–197 (2007).
[Crossref]

2006 (1)

Z. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006).
[Crossref] [PubMed]

1998 (2)

F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (1998).
[Crossref]

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. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (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]

Abdullaev, F. Kh.

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]

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(9), 093902 (2009).
[Crossref] [PubMed]

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (2012).
[Crossref]

Allayarov, I. M.

Barashenkov, I. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (2012).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

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

Bersch, C.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

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]

Cannata, F.

F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (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]

Cavalcanti, S. B.

F. C. Moreira and S. B. Cavalcanti, “Localized modes in χ(2) media with non-PT-symmetric complex localized potentials,” Phys. Rev. A (Coll. Park) 94(4), 043818 (2016).
[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(1), 013831 (2012).
[Crossref]

Christodoulides, D. N.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87(4), 043819 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

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(3), 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (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(9), 093902 (2009).
[Crossref] [PubMed]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Dai, C.

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(9), 093902 (2009).
[Crossref] [PubMed]

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(3), 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[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]

Gabadadze, G.

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017).
[Crossref]

Gao, Y.

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]

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(9), 093902 (2009).
[Crossref] [PubMed]

Hang, C.

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017).
[Crossref]

He, B.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

He, W.

He, Y.

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(15), 2723–2725 (2013).
[Crossref] [PubMed]

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(1), 013831 (2012).
[Crossref]

He, Y.-J.

Heinrich, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87(4), 043819 (2013).
[Crossref]

Hodaei, H.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Huang, G.

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017).
[Crossref]

Jiang, X.

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

Junker, G.

F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (1998).
[Crossref]

Kartashov, Y. V.

Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38(14), 2600–2603 (2013).
[Crossref] [PubMed]

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. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006).
[Crossref] [PubMed]

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]

Khajavikhan, M.

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

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(3), 192–195 (2010).
[Crossref]

Kivshar, Y. S.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (2012).
[Crossref]

Konotop, V. V.

V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39(19), 5535–5538 (2014).
[Crossref] [PubMed]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref] [PubMed]

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (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]

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), 153–197 (2007).
[Crossref]

Li, C.

Li, H.

Li, P.

Li, R.

Liu, H.

Liu, J.

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

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Malomed, B. A.

Mihalache, D.

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(1), 013831 (2012).
[Crossref]

Miri, M.-A.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87(4), 043819 (2013).
[Crossref]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

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(9), 093902 (2009).
[Crossref] [PubMed]

Moreira, F. C.

F. C. Moreira and S. B. Cavalcanti, “Localized modes in χ(2) media with non-PT-symmetric complex localized potentials,” Phys. Rev. A (Coll. Park) 94(4), 043818 (2016).
[Crossref]

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (2012).
[Crossref]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[Crossref]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
[Crossref] [PubMed]

Nixon, S.

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A (Coll. Park) 93(3), 031802 (2016).
[Crossref]

J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380(45), 3803–3809 (2016).
[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]

Nixon, S. D.

S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136(4), 459–483 (2016).
[Crossref]

Onishchukov, G.

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Peschel, U.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Regensburger, A.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

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(3), 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(9), 093902 (2009).
[Crossref] [PubMed]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6(3), 192–195 (2010).
[Crossref]

Sheng, J.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

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]

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(9), 093902 (2009).
[Crossref] [PubMed]

Sukhorukov, A. A.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (2012).
[Crossref]

Torner, L.

Z. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006).
[Crossref] [PubMed]

Trost, J.

F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (1998).
[Crossref]

Tsoy, E. N.

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(9), 093902 (2009).
[Crossref] [PubMed]

Wang, H.

Wimmer, M.

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

Xiao, M.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Xu, Z.

Z. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006).
[Crossref] [PubMed]

Yang, J.

J. Yang, “Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions,” Opt. Lett. 42(20), 4067–4070 (2017).
[Crossref] [PubMed]

S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136(4), 459–483 (2016).
[Crossref]

J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380(45), 3803–3809 (2016).
[Crossref]

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A (Coll. Park) 93(3), 031802 (2016).
[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]

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), 153–197 (2007).
[Crossref]

Yang, L.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Yulin, A. V.

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (2012).
[Crossref]

Zezyulin, D. A.

V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39(19), 5535–5538 (2014).
[Crossref] [PubMed]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref] [PubMed]

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, Y.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Zhang, Z.

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[Crossref] [PubMed]

Zheng, L.-X.

Zhu, X.

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(15), 2723–2725 (2013).
[Crossref] [PubMed]

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(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]

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
[Crossref] [PubMed]

Nat. Commun. (1)

M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, “Observation of optical solitons in PT-symmetric lattices,” Nat. Commun. 6(1), 7782 (2015).
[Crossref] [PubMed]

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(3), 192–195 (2010).
[Crossref]

Nature (1)

A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (8)

Phys. Lett. A (2)

F. Cannata, G. Junker, and J. Trost, “Schrödinger operators with complex potential but real spectrum,” Phys. Lett. A 246(3–4), 219–226 (1998).
[Crossref]

J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380(45), 3803–3809 (2016).
[Crossref]

Phys. Rev. A (9)

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, “Supersymmetry-generated complex optical potentials with real spectra,” Phys. Rev. A 87(4), 043819 (2013).
[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]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in PT-symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85(6), 063837 (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]

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]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
[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(1), 013831 (2012).
[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]

F. C. Moreira, F. K. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with PT-symmetric localized potential,” Phys. Rev. A 86(5), 053815 (2012).
[Crossref]

Phys. Rev. A (Coll. Park) (3)

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A (Coll. Park) 93(3), 031802 (2016).
[Crossref]

C. Hang, G. Gabadadze, and G. Huang, “Realization of non-PT-symmetric optical potentials with all-real spectra in a coherent atomic system,” Phys. Rev. A (Coll. Park) 95(2), 023833 (2017).
[Crossref]

F. C. Moreira and S. B. Cavalcanti, “Localized modes in χ(2) media with non-PT-symmetric complex localized potentials,” Phys. Rev. A (Coll. Park) 94(4), 043818 (2016).
[Crossref]

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

Z. Xu, Y. V. Kartashov, and L. Torner, “Stabilization of vector soliton complexes in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(5), 055601 (2006).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

Z. Zhang, Y. Zhang, J. Sheng, L. Yang, M.-A. Miri, D. N. Christodoulides, B. He, Y. Zhang, and M. Xiao, “Observation of parity-time symmetry in optically induced atomic lattices,” Phys. Rev. Lett. 117(12), 123601 (2016).
[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]

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(9), 093902 (2009).
[Crossref] [PubMed]

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

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

Science (1)

H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, “Parity-time-symmetric microring lasers,” Science 346(6212), 975–978 (2014).
[Crossref] [PubMed]

Stud. Appl. Math. (2)

S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136(4), 459–483 (2016).
[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), 153–197 (2007).
[Crossref]

Other (1)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

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Figures (6)

Fig. 1
Fig. 1 (a)-(c) are the 1D non-PT-symmetric complex potentials for c0 = 0.2, c0 = 0, and c0 = −0.4. (d)-(f) are the corresponding spectra.
Fig. 2
Fig. 2 For c0 = 0.2 and µ1 = 6. (a) and (b) are the total and partial powers of the vector solitons. (c) is the max[Re(δ)] versus the propagation constant of the second component (µ2). The profiles of the two component when µ2 = 3.9 are depicted in (d) and (e), respectively. (f) and (g) are the transverse power-flow densities of the two components. (h) and (i) exhibit the stable propagations of the two perturbed components.
Fig. 3
Fig. 3 (a) and (b) are the amplitude distributions of the two components of the vector soliton for µ2 = 4.3. (c) and (d) are the corresponding unstable propagations of the two perturbed components. The other parameters are c0 = 0.2 and µ1 = 6.
Fig. 4
Fig. 4 When c0 = 0 and µ1 = 6. (a) and (b) are the total and partial powers diagrams. (c) is the max[Re(δ)] versus µ2. (d) and (e) are the profiles of vector solitons for µ2 = 4.3 and µ2 = 4.8, respectively. (f) and (g) show the stable propagations of the two perturbed components when µ2 = 4.3. The unstable propagations of the two perturbed components for µ2 = 4.8 are depicted in (h) and (i).
Fig. 5
Fig. 5 (a)-(c) are the total power, partial power, and max[Re(δ)] versus µ2, respectively. (d) and (e) are the vector soliton profiles of µ2 = 1.65 and µ2 = 3.0, respectively. (f)-(i) are the corresponding unstable and stable propagations of the perturbed components of the two vector solitons. The other parameters are c0 = −0.4 and µ1 = 6.
Fig. 6
Fig. 6 For c0 = −0.4 and µ1 = 6. (a)-(c) are the profile of the vector soliton and the unstable propagations of the two perturbed components when µ2 = 3.43. The vector soliton profile and the stable propagations of the two perturbed components for µ2 = 3.6 are depicted in (d)-(f). (g) is the vector soliton profile when µ2 = 3.9. (h) and (i) show the corresponding unstable propagations of the two perturbed components.

Equations (7)

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i U 1,2 z + 2 U 1,2 x 2 +V(x) U 1,2 +( | U 1 | 2 + | U 2 | 2 ) U 1,2 =0.
V(x)= g 2 (x)+2 c 0 g(x)+i g x (x).
g(x)=tanh2(x+2)tanh(x2).
[ 2 x 2 +V(x)]ψ=λψ.
2 q 1,2 x 2 +V(x) q 1,2 +( | q 1 | 2 + | q 2 | 2 ) q 1,2 μ 1,2 q 1,2 =0.
U 1,2 (x,z)= e i μ 1,2 z [ q 1,2 (x)+ g 1,2 (x) e δz + t 1,2 * (x) e δ * z ].
{ g 1 δ=i[( μ 1 + 2 x 2 +V+2 | q 1 | 2 + | q 2 | 2 ) g 1 + q 1 2 t 1 + q 1 q 2 * g 2 + q 1 q 1 t 2 ], t 1 δ=i[ ( q 1 2 ) * g 1 +( μ 1 2 x 2 V * 2 | q 1 | 2 | q 2 | 2 ) t 1 q 1 * q 2 * g 2 q 1 * q 2 t 2 ], g 2 δ=i[ q 1 * q 2 g 1 + q 1 q 2 t 1 +( μ 2 + 2 x 2 +V+ | q 1 | 2 +2 | q 2 | 2 ) g 2 + q 2 2 t 2 ], t 2 δ=i[ q 1 * q 2 * g 1 q 1 q 2 * t 1 ( q 2 2 ) * g 2 +( μ 2 2 x 2 V * | q 1 | 2 2 | q 2 | 2 ) t 2 ].

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