Abstract

The existence and stability of gap solitons are investigated in the semi-infinite gap of a parity-time (PT)-symmetric periodic potential (optical lattice) with a higher-order diffraction. The Bloch bands and band gaps of this PT-symmetric optical lattice depend crucially on the coupling constant of the fourth-order diffraction, whereas the phase transition point of this PT optical lattice remains unchangeable. The fourth-order diffraction plays a significant role in destabilizing the propagation of dipole solitons. Specifically, when the fourth-order diffraction coupling constant increases, the stable region of the dipole solitons shrinks as new regions of instability appear. However, fundamental solitons are found to be always linearly stable with arbitrary positive value of the coupling constant. We also investigate nonlinear evolution of the PT solitons under perturbation.

© 2014 Optical Society of America

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  3. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).
  4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical PT-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
  5. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
  6. C. E. Rueter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
  7. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
  8. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
  9. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z.H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
  10. Y. Meng and Y. Liu, “Power-dependent shaping of solitons in parity time symmetric potentials with spatially modulated nonlinearity,” J. Opt. Soc. Am. B 30, 1148–1153 (2013).
  11. A. K. Sarma, “Modulation instability in nonlinear complex parity-time symmetric periodic structures,” J. Opt. Soc. Am. B 31, 1861–1866 (2014).
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  17. 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, 013808 (2012).
  18. H. Li, X. Zhu, Z. Shi, B. A. Malomed, T. Lai, and C. Lee, “Bulk vortices and half-vortex surface modes in parity-time-symmetric media,” Phys. Rev. A 89, 053811 (2014)
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  20. Y. V. Kartashov, “Vector solitons in parity-time-symmetric lattices,” Opt. Lett. 38, 2600–2603 (2013).
  21. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).
  22. Z. Lu and Z. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).
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  25. F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).
  26. H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
  27. C. Yin, Y. He, H. Li, and J. Xie, “Solitons in parity-time symmetric potentials with spatially modulated nonlocal nonlinearity,” Opt. Express 20, 19355–19362 (2012).
  28. X. Zhu, H. Li, H. Wang, and Y. He, “Nonlocal multihump solitons in parity-time symmetric periodic potentials,” J. Opt. Soc. Am. B 30, 1987–1995 (2013).
  29. C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).
  30. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).
  31. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
  32. G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).
  33. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
  34. W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).
  35. S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).
  36. L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
  37. M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–308 (2010).
  38. G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).
  39. J. T. Cole and Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. A 90, 013815 (2014).
  40. K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).
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    [Crossref]

2014 (5)

A. K. Sarma, “Modulation instability in nonlinear complex parity-time symmetric periodic structures,” J. Opt. Soc. Am. B 31, 1861–1866 (2014).

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

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).

J. T. Cole and Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. A 90, 013815 (2014).

2013 (7)

G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).

Y. Meng and Y. Liu, “Power-dependent shaping of solitons in parity time symmetric potentials with spatially modulated nonlinearity,” J. Opt. Soc. Am. B 30, 1148–1153 (2013).

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

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

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).

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

2012 (8)

J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

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

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, 013808 (2012).

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

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

C. Yin, Y. He, H. Li, and J. Xie, “Solitons in parity-time symmetric potentials with spatially modulated nonlocal nonlinearity,” Opt. Express 20, 19355–19362 (2012).

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

2011 (7)

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

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

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

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3302 (2011).

H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).

Z. Lu and Z. Zhang, “Defect solitons in parity-time symmetric superlattices,” Opt. Express 19, 11457–11462 (2011).

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

2010 (2)

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

M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–308 (2010).

2009 (4)

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).

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

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, 030402 (2008).

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

2007 (1)

2006 (1)

K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).

2005 (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).

2002 (1)

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

1998 (1)

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

Abdullaev, F. K.

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

Aceves, A. B.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

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, 013808 (2012).

Agrawal, G. P.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).

Aimez, V.

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

Akylas, T.R.

T.R. Akylas, G. Hwang, and J. Yang, “From nonlocal gap solitary waves to bound states in periodic media,” Proc. Roy. Soc. A, doi: (2011).
[Crossref]

Alberucci, A.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

Assanto, G.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

Bender, C. M.

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

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

Bersch, C.

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

Bhadra, S. K.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).

Boettcher, S.

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

Brazhnyi, V. A.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

Brody, D. C.

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

Burlak, G.

G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).

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, 013808 (2012).

Chen, Z.

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

Christodoulides, D. N.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

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

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

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

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

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

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

Cole, J. T.

J. T. Cole and Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. A 90, 013815 (2014).

Colet, P.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

Danckaert, J.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

de Valcarcel, G. J.

K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).

Driben, R.

Duchesne, D.

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

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, 1019–1041 (2011).

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

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

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

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, 013808 (2012).

Ge, L.

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

Gelens, L.

M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–308 (2010).

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

Gomila, D.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

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, 093902 (2009).

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, 2723–2725 (2013).

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

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).

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

C. Yin, Y. He, H. Li, and J. Xie, “Solitons in parity-time symmetric potentials with spatially modulated nonlocal nonlinearity,” Opt. Express 20, 19355–19362 (2012).

Herrero, R.

K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).

Hwang, G.

T.R. Akylas, G. Hwang, and J. Yang, “From nonlocal gap solitary waves to bound states in periodic media,” Proc. Roy. Soc. A, doi: (2011).
[Crossref]

Jiang, X.

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

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3302 (2011).

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

Jisha, C. P.

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

Jones, H. F.

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

Kartashov, Y. V.

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, 013808 (2012).

Kip, D.

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

Konotop, V. V.

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).

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

Kottos, T.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

Kovanis, V.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

Kozyreff, G.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Lai, T.

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

Lan, Y.

J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).

Lee, C.

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

Li, H.

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

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

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

C. Yin, Y. He, H. Li, and J. Xie, “Solitons in parity-time symmetric potentials with spatially modulated nonlocal nonlinearity,” Opt. Express 20, 19355–19362 (2012).

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

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3302 (2011).

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

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

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, 013831 (2012).

Liu, W.

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

Liu, Y.

Louvergneaux, E.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Lu, Z.

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, 1019–1041 (2011).

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

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

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

Malomed, B. A.

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

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).

G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).

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

Matias, M. A.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

Meng, Y.

Mihalache, D.

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).

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

Miri, M. A.

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

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

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, 093902 (2009).

Moreira, F. C.

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).

Musslimani, Z. H.

J. T. Cole and Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. A 90, 013815 (2014).

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

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

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, 1019–1041 (2011).

Mussot, A.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Nixon, S.

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

Onishchukov, G.

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

Peschel, U.

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

Regensburger, A.

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

Roy, S.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).

Rueter, C. E.

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).

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, 093902 (2009).

Sarma, A. K.

Segev, M.

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

Shi, Z.

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

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

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3302 (2011).

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

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, 093902 (2009).

Staliunas, K.

K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).

Taki, M.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Tian, B.

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

Tlidi, M.

M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Opt. Lett. 35, 306–308 (2010).

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Torner, L.

Van der Sande, G.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

Vladimirov, A. G.

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

Volatier-Ravat, M.

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

Wang, H.

Wang, J.

Xie, J.

Xu, T.

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

Yang, J.

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

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

T.R. Akylas, G. Hwang, and J. Yang, “From nonlocal gap solitary waves to bound states in periodic media,” Proc. Roy. Soc. A, doi: (2011).
[Crossref]

Yin, C.

Zeng, J.

J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).

Zezyulin, D. A.

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

Zhang, H.

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

Zhang, Z.

Zhu, X.

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

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

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

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

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

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

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3302 (2011).

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, 1019–1041 (2011).

J. Opt. Soc. Am. B (3)

J. Phys. A: Math. Gen. (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A: Math. Gen. 38, 171–176 (2005).

Nature (1)

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

Nature Phys. (1)

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

Opt. Express (3)

Opt. Lett. (7)

Phys. Rev. A (14)

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).

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, 013808 (2012).

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

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

W. Liu, B. Tian, H. Zhang, T. Xu, and H. Li, “Solitary wave pulses in optical fibers with normal dispersion and higher-order effects,” Phys. Rev. A 79, 063810 (2009).

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79, 023824 (2009).

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matias, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).

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

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

F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in PT-symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).

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

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

C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014).

J. T. Cole and Z. H. Musslimani, “Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. A 90, 013815 (2014).

Phys. Rev. A. (1)

M. A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, “Bragg solitons in nonlinear PT-symmetric periodic potentials,” Phys. Rev. A. 86, 033801 (2012).

Phys. Rev. E (3)

J. Zeng and Y. Lan, “Two-dimensional solitons in PT linear lattice potentials,” Phys. Rev. E 85, 047601 (2012).

G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).

K. Staliunas, R. Herrero, and G. J. de Valcarcel, “Subdiffractive band-edge solitons in Bose-Einstein condensates in periodic potentials,” Phys. Rev. E 73, 065603(R) (2006).

Phys. Rev. Lett. (5)

G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki, and A. G. Vladimirov, “Localized beating between dynamically generated frequencies,” Phys. Rev. Lett. 102, 043905 (2009).

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

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

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

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

Other (2)

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

T.R. Akylas, G. Hwang, and J. Yang, “From nonlocal gap solitary waves to bound states in periodic media,” Proc. Roy. Soc. A, doi: (2011).
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Diffraction relations of PT lattices with a fourth-order diffraction (top) for three β values 0 (top left), 0.25 (top middle) and 0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.6 (bottom middle and right) at V0 = 6 and β = 0.25.
Fig. 2
Fig. 2 Band-gap structure of PT lattices with a fourth-order diffraction (left) as increasing the β value from zero to one at V0 = 6 and W0 = 0.45, (right) as W0 crosses the phase transition point with β = 0.25 and V0 = 6.
Fig. 3
Fig. 3 Diffraction relations of PT lattices with a fourth-order diffraction (top) for three negative β values −0.1 (top left), −0.25 (top middle) and −0.5 (top right) at V0 = 6 and W0 = 0.45, and (bottom) for W0 = 0.5 (bottom left) and W0 = 0.51 (bottom middle and right) at V0 = 6 and β = −0.25.
Fig. 4
Fig. 4 Power curves of PT solitons in the semi-infinite gap under focusing cubic nonlinearity (σ = 1) for three β values 0 (left), 0.25 (middle) and 0.5 (right) at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons. Solid blue and dashed red lines represent stable and unstable solitons, respectively (the same holds for all other figures).
Fig. 5
Fig. 5 Fundamental solitons (top left) and dipole solitons (top middle and right) and their linear-stability spectra (bottom) for three different μ values −3.5 (left), −3.9 (middle) and −3.3 (right) in the semi-infinite gap at V0 = 6, W0 = 0.45 and β = 0.5. The solid blue lines are for the real part and dashed pink lines for the imaginary part. The power curves of these solitons are shown in Fig. 4 (right), and the locations of these solitons are marked by dots on that power curves.
Fig. 6
Fig. 6 Power curves of fundamental and dipole solitons for three μ values −4.5 (left), −4 (middle) and −3.5 (right) as increasing the fourth-order coupling constant from zero to 1 at V0 = 6 and W0 = 0.45. The lower curves are for fundamental solitons and the upper curves for dipole solitons.
Fig. 7
Fig. 7 Dipole solitons (top) and their linear-stability spectra (bottom) for three μ values −4.5 (left), −4 (middle) and −3.5 (right) in the semi-infinite gap when β = 0.5 (left and middle) and β = 0.95 (right). The power curves of these dipole solitons are shown in Fig. 6, and the locations of these solitons are marked by dots on that power curve.
Fig. 8
Fig. 8 Nonlinear evolution of the linearly stable fundamental solitons with μ = −3.5 (left) shown in Fig. 5 (left) and the linearly stable dipole solitons with μ = −4.5 (right) shown in Fig. 7 (left) at β = 0.5 under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane.
Fig. 9
Fig. 9 Nonlinear evolution of the linearly unstable dipole solitons shown in Fig. 7 (middle and right) under 10% random noise perturbations. Shown is the field |U(x, z)| in the (x, z) plane.

Equations (13)

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i U z + U x x β U x x x x + V ( x ) U + σ | U | 2 U = 0 ,
V ( x ) = V 0 [ cos 2 ( x ) + i W 0 sin ( 2 x ) ] .
i U z + U x x β U x x x x + V ( x ) U = 0 ,
U ( x , z ) = p ( x ; k ) e i k x i μ z ,
U ( x , z ) = u ( x ) e i μ z ,
U = e i μ t [ u ( x ) + f ( x ) e λ z + g * ( x ) e λ * z ] ,
i ( f g ) = λ ( f g ) ,
= ( L 11 L 12 L 21 L 22 ) ,
L 11 = μ + x x β x x x x + V ( x ) + 2 σ | u | 2 ,
L 12 = σ u 2 ,
L 21 = σ ( u 2 ) * ,
L 22 = [ μ + x x β x x x x + V * ( x ) + 2 σ | u | 2 ] .
P ( μ ) = | u ( x ; μ ) | 2 d x .

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