Abstract

We investigate the existence, stability and internal interaction of two-dimensional multipole solitons in defocusing PT symmetric nonlocal nonlinear media. Compared with nonlocal fundamental solitons in PT-symmetric lattices, the multipole solitons reveal the novel internal interaction such as the oscillation of the gravity center. We consider the change of the gravity center of dipole solitons during propagation, and find that these dipole solitons have “breather-like” or “competition” behavior in different nonlinearities. We also demonstrate that dipole solitons are easier to be stabilized under the intermediate nonlocality. Moreover, two more complicated multipole solitons are studied and it is found that there also exist the novel internal interaction for the multipole solitons.

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

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2017 (1)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

2016 (6)

Y. He, X. Zhu, and D. Mihalache, “Dynamics of spatial solitons in parity-time-symmetric optical lattices: A selection of recent theoretical results,” Rom. J. Phys. 61, 595–613 (2016).

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
[Crossref]

H. Wang, Z. Li, X. Ren, and Y. Weng, “Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity,” Opt. Express 24(20), 23063–23071 (2016).
[Crossref] [PubMed]

A. Kotb and K. Zoiros, “Soliton all-optical logic AND gate with semiconductor optical amplifier-assisted Mach–Zehnder interferometer,” Opt. Eng. 55(8), 087109 (2016).
[Crossref]

A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016).
[Crossref]

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” J. Opt. 18(12), 125501 (2016).
[Crossref]

2015 (4)

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

X. Zhu, Z. Shi, and H. Li, “Nonlocal gray solitons in parity-time-symmetric potentials with spatially modulated nonlinearity,” Opt. Commun. 355, 516–522 (2015).
[Crossref]

M. Li and T. Xu, “Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(3), 033202 (2015).
[Crossref] [PubMed]

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).

2014 (1)

2013 (3)

2012 (3)

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (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]

2011 (2)

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

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 (2)

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]

2007 (2)

2006 (4)

2005 (3)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30(4), 415–417 (2005).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72(5), 698–704 (2005).
[Crossref]

2004 (1)

G. Assanto, M. Peccianti, and C. Conti, “One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 10(5), 862–869 (2004).
[Crossref]

2001 (1)

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1), 016612 (2001).
[Crossref] [PubMed]

1998 (1)

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998).
[Crossref]

1997 (2)

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[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]

Alberucci, A.

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” J. Opt. 18(12), 125501 (2016).
[Crossref]

Anderson, D. Z.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Assanto, G.

A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016).
[Crossref]

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” J. Opt. 18(12), 125501 (2016).
[Crossref]

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30(4), 415–417 (2005).
[Crossref] [PubMed]

G. Assanto, M. Peccianti, and C. Conti, “One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 10(5), 862–869 (2004).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Bang, O.

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1), 016612 (2001).
[Crossref] [PubMed]

Bender, C. M.

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

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Chen, L.

Chen, W.

Chen, Y. F.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Christodoulides, D. N.

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]

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]

Cohen, O.

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31(22), 3312–3314 (2006).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Conti, C.

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30(4), 415–417 (2005).
[Crossref] [PubMed]

G. Assanto, M. Peccianti, and C. Conti, “One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 10(5), 862–869 (2004).
[Crossref]

Dai, L.

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

Denz, C.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[Crossref] [PubMed]

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998).
[Crossref]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

El-Ganainy, R.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

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]

Fainman, Y.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Feng, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Fischer, R.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[Crossref] [PubMed]

Ge, L.

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

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]

Guo, Q.

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

He, Y.

Y. He, X. Zhu, and D. Mihalache, “Dynamics of spatial solitons in parity-time-symmetric optical lattices: A selection of recent theoretical results,” Rom. J. Phys. 61, 595–613 (2016).

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
[Crossref]

Hu, S.

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

Hu, W.

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

Huang, J.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Jeng, C.-C.

Jiang, X.

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (2012).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Jisha, C. P.

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” J. Opt. 18(12), 125501 (2016).
[Crossref]

Kartashov, Y. V.

Kivshar, Y. S.

C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. 32(4), 397–399 (2007).
[Crossref] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[Crossref] [PubMed]

Kong, Q.

Konotop, V. V.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
[Crossref]

Kotb, A.

A. Kotb and K. Zoiros, “Soliton all-optical logic AND gate with semiconductor optical amplifier-assisted Mach–Zehnder interferometer,” Opt. Eng. 55(8), 087109 (2016).
[Crossref]

Krolikowski, W.

Królikowski, W.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998).
[Crossref]

Lee, R.-K.

Li, H.

X. Zhu, Z. Shi, and H. Li, “Nonlocal gray solitons in parity-time-symmetric potentials with spatially modulated nonlinearity,” Opt. Commun. 355, 516–522 (2015).
[Crossref]

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
[Crossref]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (2012).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Li, M.

M. Li and T. Xu, “Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(3), 033202 (2015).
[Crossref] [PubMed]

Li, Z.

Lin, Y.-Y.

Lu, D.

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

Lu, M. H.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Luther-Davies, B.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998).
[Crossref]

Ma, C.

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

Ma, X.

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

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. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72(5), 698–704 (2005).
[Crossref]

Mamaev, A. V.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Mezentsev, V. K.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Mihalache, D.

Y. He, X. Zhu, and D. Mihalache, “Dynamics of spatial solitons in parity-time-symmetric optical lattices: A selection of recent theoretical results,” Rom. J. Phys. 61, 595–613 (2016).

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

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

Neshev, D. N.

C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. 32(4), 397–399 (2007).
[Crossref] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[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]

Peccianti, M.

M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30(4), 415–417 (2005).
[Crossref] [PubMed]

G. Assanto, M. Peccianti, and C. Conti, “One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 10(5), 862–869 (2004).
[Crossref]

Piccardi, A.

A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016).
[Crossref]

Rasmussen, J. J.

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1), 016612 (2001).
[Crossref] [PubMed]

Ren, X.

Residori, S.

A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016).
[Crossref]

Rosberg, C. R.

Rotschild, C.

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31(22), 3312–3314 (2006).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Saffman, M.

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (1998).
[Crossref]

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72(5), 698–704 (2005).
[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]

Scherer, A.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Segev, M.

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31(22), 3312–3314 (2006).
[Crossref] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

Shen, M.

Shi, J.

Shi, Z.

X. Zhu, Z. Shi, and H. Li, “Nonlocal gray solitons in parity-time-symmetric potentials with spatially modulated nonlinearity,” Opt. Commun. 355, 516–522 (2015).
[Crossref]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (2012).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

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]

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Sukhorukov, A. A.

C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. 32(4), 397–399 (2007).
[Crossref] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[Crossref] [PubMed]

Torner, L.

Träger, D.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[Crossref] [PubMed]

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]

Vysloukh, V. A.

Wang, H.

Wang, Q.

Weng, Y.

Wu, Y. D.

Wyller, J.

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1), 016612 (2001).
[Crossref] [PubMed]

Xu, T.

M. Li and T. Xu, “Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(3), 033202 (2015).
[Crossref] [PubMed]

Xu, Y. L.

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

Xu, Z.

Yang, J.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (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]

Zang, T.

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

Zezyulin, D. A.

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
[Crossref]

Zhao, H.

Zhu, X.

Y. He, X. Zhu, and D. Mihalache, “Dynamics of spatial solitons in parity-time-symmetric optical lattices: A selection of recent theoretical results,” Rom. J. Phys. 61, 595–613 (2016).

X. Zhu, Z. Shi, and H. Li, “Nonlocal gray solitons in parity-time-symmetric potentials with spatially modulated nonlinearity,” Opt. Commun. 355, 516–522 (2015).
[Crossref]

X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30(7), 1987–1995 (2013).
[Crossref]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (2012).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Zoiros, K.

A. Kotb and K. Zoiros, “Soliton all-optical logic AND gate with semiconductor optical amplifier-assisted Mach–Zehnder interferometer,” Opt. Eng. 55(8), 087109 (2016).
[Crossref]

Zozulya, A. A.

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Europhys. Lett. (3)

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Europhys. Lett. 98(6), 64006–64011 (2012).
[Crossref]

S. Hu, D. Lu, X. Ma, Q. Guo, and W. Hu, “Defect solitons supported by nonlocal PT symmetric superlattices,” Europhys. Lett. 98(1), 14006–14011 (2012).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Higher-order vortex solitons, multipoles, and supervortices on a square optical lattice,” Europhys. Lett. 72(5), 698–704 (2005).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

G. Assanto, M. Peccianti, and C. Conti, “One-dimensional transverse modulational instability in nonlocal media with a reorientational nonlinearity,” IEEE J. Sel. Top. Quantum Electron. 10(5), 862–869 (2004).
[Crossref]

J. Opt. (2)

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” J. Opt. 18(12), 125501 (2016).
[Crossref]

A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016).
[Crossref]

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

Nat. Photonics (1)

L. Feng, R. El-Ganainy, and L. Ge, “Non-Hermitian photonics based on parity–time symmetry,” Nat. Photonics 11(12), 752–762 (2017).
[Crossref]

Opt. Commun. (1)

X. Zhu, Z. Shi, and H. Li, “Nonlocal gray solitons in parity-time-symmetric potentials with spatially modulated nonlinearity,” Opt. Commun. 355, 516–522 (2015).
[Crossref]

Opt. Eng. (1)

A. Kotb and K. Zoiros, “Soliton all-optical logic AND gate with semiconductor optical amplifier-assisted Mach–Zehnder interferometer,” Opt. Eng. 55(8), 087109 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (8)

Phys. Rev. A (2)

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

A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56(2), 1110–1113 (1997).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (3)

M. Li and T. Xu, “Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(3), 033202 (2015).
[Crossref] [PubMed]

L. Ge, M. Shen, T. Zang, C. Ma, and L. Dai, “Stability of optical solitons in parity-time-symmetric optical lattices with competing cubic and quintic nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 91(2), 023203 (2015).
[Crossref] [PubMed]

W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(1), 016612 (2001).
[Crossref] [PubMed]

Phys. Rev. Lett. (6)

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96(2), 023905 (2006).
[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]

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]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95(21), 213904 (2005).
[Crossref] [PubMed]

W. Królikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous interaction of spatial solitons in photorefractive media,” Phys. Rev. Lett. 80(15), 3240–3243 (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]

Rep. Prog. Phys. (1)

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

Rev. Mod. Phys. (1)

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
[Crossref]

Rom. J. Phys. (1)

Y. He, X. Zhu, and D. Mihalache, “Dynamics of spatial solitons in parity-time-symmetric optical lattices: A selection of recent theoretical results,” Rom. J. Phys. 61, 595–613 (2016).

Rom. Rep. Phys. (1)

D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).

Science (2)

L. Feng, M. Ayache, J. Huang, Y. L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333(6043), 729–733 (2011).
[Crossref] [PubMed]

A. W. Snyder and D. J. Mitchell, “Accessible Solitons,” Science 276(5318), 1538–1541 (1997).
[Crossref]

Other (2)

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (John Wiley & Sons, Inc., 2011), Chap. 7.

J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010), Chap. 7.

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Figures (9)

Fig. 1
Fig. 1 The profiles of the refractive index of (a) the weak nonlocality, (b) the intermediate nonlocality and (c) the strong nonlocality.
Fig. 2
Fig. 2 The power curves of the dipole solitons with different nonlocalities in the first gap. The blue lines indicate the in-phase dipole solitons and the red lines indicate the out-of-phase dipole solitons. The dash lines represent the unstable solitons and the green solid lines represent the stable solitons. The gray areas represent the energy bands. σ 1 =1, W 0 0.3 , V 0 =8.
Fig. 3
Fig. 3 The out-of-phase dipole solitons in a PT symmetric weak-nonlocal medium with μ=9.1, d=0.05, corresponding to point A in Fig. 2. (a), (d) and (e) are the profiles of the dipole soliton at z = 0, 100 and 160. (b) is the imaginary part of the dipole soliton. (c) is the change of the gravity center during propagation. (f) is the spectrum of perturbation growth rate.
Fig. 4
Fig. 4 The in-phase dipole solitons in a PT symmetric medium with intermediate nonlocality ( d=0.5). The upper row represents the stable dipole soliton with μ=9.2, corresponding to point B in Fig. 2. The lower row represents the unstable dipole soliton with μ=8.4, corresponding to point C in Fig. 2. (a), (b), (e) and (f) are the profiles of the corresponding dipole solitons. (c) and (g) are the spectrums of perturbation growth rate. (d) is the change of the gravity center of the dipole soliton with μ=9.2 during propagation. (h) is the imaginary part of the in-phase solitons.
Fig. 5
Fig. 5 The in-phase soliton in a PT symmetric strong-nonlocal medium with μ=9.4, d=3. (a) and (b) is the profile of the dipole soliton at z = 40 and 80. (c) is the spectrum of perturbation growth rate. (d) is the change of the gravity center of the dipole soliton during propagation.
Fig. 6
Fig. 6 The existence and stable range of the in-phase solitons with variable degree of nonlocality ( σ=1, V 0 =8, W 0 =0.3). The gray areas indicate the energy bands. The area inside the blue lines is the existence range and the green area is the stable range. The difference between different nonlocalities is about one order of magnitude. The x-axis is a logarithmic coordinate.
Fig. 7
Fig. 7 The existence and stability of the in-phase solitons with varied gain-loss components. (a) is the weak nonlocality case. (b) is the intermediate nonlocality case. (c) is the strong nonlocality case. The gray areas is the energy bands. The upper energy band and the lower energy band merges when W 0 =0.5. The area inside the blue lines represents the existence range and the green area represents the stable range.
Fig. 8
Fig. 8 The quadrupole solitons in a PT symmetric nonlocal medium with (a-e) d=0.5, μ=9, , (f) d=3, μ=9.65. (a) and (b) are the profile of the quadrupole soliton with intermediate nonlocality at z = 0 and 290. (c) is the change of the gravity center of the quadrupole soliton with intermediate nonlocality. (d) is the imaginary part of quadrupole solitons. (e) is the phase structure of quadrupole solitons. (f) is the change of the gravity center of the quadrupole soliton with strong nonlocality.
Fig. 9
Fig. 9 The 8-hump solitons in a PT symmetric intermediate-nonlocal medium with d=0.5, μ=9. (a), (b) and (c) are the intensity profile of the 8-hump soliton at z = 0, 50 and 100. (d) is the change of the gravity center of the 8-hump soliton.

Equations (6)

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I U z + U+(V+iW)U+σnU=0 d nn+ | U | 2 =0
n(I)= + + R(x x ,y y ) I( x y )d x d y
R(x,y)= 1 2π d exp( x 2 + y 2 d 2 )
{ μq+ q+(V+iW)q+σnq=0 d nn+ | q | 2 =0
U(x,y,z)=[ q(x,y)+F(x,y) e λz + G * (x,y) e λ * z ]
{ λF=i[ μF+ F+(V+iW)F+σnG+σqΔn ] λG=i[ μG G(V+iW)G+σnFσ q * Δn ]

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