Abstract

We investigate the evolution of two-dimensional solitons in PT-symmetric optical lattices with defocusing nonlocal nonlinearity. We discuss the existence of 2D fundamental solitons, and we numerically stimulate the propagation of these solitons in the optical lattice. We also discuss how different degrees of nonlocality can affect the evolution of solitons, and discover that nonlocality can have significant impacts on the formation and propagation of the solitons. In addition, we find that PT-symmetry can lead to fission and shifting of solitons, and this phenomenon is also affected by the degree of nonlocality.

© 2016 Optical Society of America

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References

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  2. C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
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  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]
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  5. 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).
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  6. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
    [Crossref] [PubMed]
  7. A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
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  8. Y. A. Zahznyak and A. I. Yakimenko, “Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates,” Phys. Lett. A 372(16), 2862–2866 (2008).
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  10. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
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  12. N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
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    [Crossref] [PubMed]
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  22. D. Mihalache, “Localized structures in nonlinear optical media: A selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).
  23. 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]
  24. X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
    [Crossref]
  25. H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
    [Crossref]
  26. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
    [Crossref]
  27. Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Eur. Phys. Lett 98(6), 64006 (2012).
    [Crossref]
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2016 (2)

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

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).

2015 (2)

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

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

2013 (3)

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[Crossref] [PubMed]

2012 (4)

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (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]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defocusing Kerr media supported by PT symmetric potentials,” Eur. Phys. Lett 98(6), 64006 (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(1), 013831 (2012).
[Crossref]

2011 (2)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[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]

2010 (2)

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]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, “Binary parity-time-symmetric nonlinear lattices with balanced gain and loss,” Opt. Lett. 35(17), 2976–2978 (2010).
[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)

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]

Y. A. Zahznyak and A. I. Yakimenko, “Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates,” Phys. Lett. A 372(16), 2862–2866 (2008).
[Crossref]

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]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
[Crossref] [PubMed]

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46(4), 277–292 (2005).
[Crossref]

2004 (1)

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
[Crossref] [PubMed]

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

2002 (1)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[Crossref] [PubMed]

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)

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

1993 (1)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref] [PubMed]

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]

Alù, A.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
[Crossref] [PubMed]

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.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[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]

Bender, C. M.

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46(4), 277–292 (2005).
[Crossref]

Bender, N.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

Benderv, C. M.

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

Blasberg, T.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref] [PubMed]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

Boettcher, S.

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

Castaldi, G.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

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]

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.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[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]

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. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Cohen, 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]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
[Crossref] [PubMed]

Dmitriev, S. V.

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.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[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]

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]

Ellis, F. M.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

Engheta, N.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

Factor, S.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[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, 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]

Galdi, V.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

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]

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).

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]

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]

Jiang, X.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

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

Kip, D.

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

Kivshar, Y.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
[Crossref] [PubMed]

Kivshar, Y. S.

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]

Kottos, T.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

Krolikowski, W.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[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]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[Crossref] [PubMed]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Li, H.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

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

Li, Z.

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

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]

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]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[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]

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]

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]

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).

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]

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.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[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]

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.

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
[Crossref] [PubMed]

Ramezani, H.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

Rasmussen, J. J.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[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]

Ren, X.

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

Rotschild, C.

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]

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]

Savoia, S.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[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. 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]

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]

Shi, Z.

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

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[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]

Sukhorukov, A. A.

Suter, D.

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref] [PubMed]

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[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]

Wang, H.

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

Wang, H. C.

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

Wyller, J.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[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]

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]

Yakimenko, A. I.

Y. A. Zahznyak and A. I. Yakimenko, “Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates,” Phys. Lett. A 372(16), 2862–2866 (2008).
[Crossref]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
[Crossref] [PubMed]

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]

Zahznyak, Y. A.

Y. A. Zahznyak and A. I. Yakimenko, “Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates,” Phys. Lett. A 372(16), 2862–2866 (2008).
[Crossref]

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
[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]

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).

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]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (2012).
[Crossref]

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

Contemp. Phys. (1)

C. M. Bender, “Introduction to PT-symmetric quantum theory,” Contemp. Phys. 46(4), 277–292 (2005).
[Crossref]

Eur. Phys. Lett (1)

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

Int. J. Theor. Phys. (1)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50(4), 1019–1041 (2011).
[Crossref]

Nat. Phys. (1)

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

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

X. Ren, H. Wang, Z. Li, and H. C. Wang, “Solitons supported by two-dimensional mixed linear-nonlinear complex optical lattices,” Opt. Commun. 356, 230–235 (2015).
[Crossref]

Opt. Lett. (1)

Phys. Lett. A (1)

Y. A. Zahznyak and A. I. Yakimenko, “Three-dimensional solitons and vortices in dipolar Bose-Einstein condensates,” Phys. Lett. A 372(16), 2862–2866 (2008).
[Crossref]

Phys. Rev. A (4)

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).
[Crossref] [PubMed]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86(2), 023840 (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]

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]

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

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]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046619 (2002).
[Crossref] [PubMed]

A. I. Yakimenko, Y. A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(6), 065603 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (8)

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

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref] [PubMed]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110(17), 173901 (2013).
[Crossref] [PubMed]

N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
[Crossref] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92(11), 113902 (2004).
[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]

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]

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 (1)

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]

Other (1)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (Society for Industrial and Applied Mathematics, 2010).

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Figures (9)

Fig. 1
Fig. 1 The band structure of the PT-symmetric optical lattice for V 0 =10 and W 0 =0.1 , zone 1 is the semi-infinite band gap and zone 2 is the first band gap.
Fig. 2
Fig. 2 The corresponding diagrams of soliton and nonlocal response profiles with degree of nonlocality (a) d=0.01 (weak-nonlocal), (b) d=0.5 (general-nonlocal), (c) d=5 (strong-nonlocal). The blue lines represent soliton profiles and the red lines represent nonlocal response profiles.
Fig. 3
Fig. 3 (a), (b), (c) are the profiles of the corresponding refractive index of the different degree of nonlocality [Figs. 2(a), 2(b), 2(c)], respectively.
Fig. 4
Fig. 4 The power diagrams for d=0.01 (yellow lines), d=0.5 (green lines), d=5 (blue lines) with (a) W 0 =0.1 ,the solid and dashed lines indicate stable and unstable regions, respectively. (b) W 0 =0 , red lines indicate the solitons are unstable and blue, green and yellow lines indicate the solitons are stable. The blue shaded regions are the Bloch bands.
Fig. 5
Fig. 5 (a) is the soliton profile for μ=6.1 at z=0 , (b) is the soliton for μ=6.1 at z=400 , (c) indicates the growth rate for μ=6.1 , and (d) is the soliton profile for μ=8.6 at z=0 , (e) is the soliton for μ=8.6 at z=400 , (f) indicates the growth rate for μ=8.6 .
Fig. 6
Fig. 6 (a) is the soliton profile for μ=6.4 at z=0 , (b) is the soliton for μ=6.4 at z=400 , (c) indicates the growth rate for μ=6.4 , and (d) is the soliton profile for μ=8.4 at z=0 , (e) is the soliton for μ=8.4 at z=400 , (f) indicates the growth rate for μ=-8.4 .
Fig. 7
Fig. 7 (a) is the soliton profile for μ=6.2 at z=0 , (b) is the soliton for μ=6.2 at z=400 , (c) indicates the growth rate for μ=6.2 , and (d) is the soliton profile for μ=7.2 at z=0 , (e) is the soliton for μ=7.2 at z=400 , (f) indicates the growth rate for μ=7.2 .
Fig. 8
Fig. 8 (a) shows the stable region of solitons with the change of degree of nonlocality, and (b) shows the stable region of solitons with the change of W 0 , the shaded region represents stable region.
Fig. 9
Fig. 9 The first column are the soliton profiles of μ=7 for d=1,5,10 at z=200 , respectively, the second column are the soliton profiles of μ=7 for d=1,5,10 at z=300 , respectively, the third column are the soliton profiles of μ=7 for d=1,5,10 at z=400 , respectively.

Equations (6)

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

i U z + 2 U x 2 + 2 U y 2 +V(x,y)U+σnU=0, d 2 n x 2 n+|U | 2 =0,
μF( 2 x 2 +2i k x x k x 2 )F( 2 y 2 +2i k y y k y 2 )FVF=0
μu+( 2 x 2 + 2 y 2 )u+Vu+σnu=0 d( 2 x 2 + 2 y 2 )un+|u | 2 =0
U(x,y,z)= e iμz [u(x,y)+F(x,y) e λz + G * (x,y) e λ * z ]
λF=i[μF+( 2 x 2 + 2 x 2 )F+(V+iW)F+σnF+σuΔn] λG=i[μG( 2 x 2 + 2 x 2 )G(ViW)GσnFσ u * Δn]
n= g(x x ' ,y y ' )|q( x ' , y ' ) | 2 d x ' d y ' Δn= g(x x ' ,y y ' )[u( x ' , y ' )G( x ' , y ' )+ u * ( x ' , y ' )F( x ' , y ' )] d x ' d y ' g(x,y)=(1/ 2π d )exp( x 2 + y 2 / d )

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