Abstract

The symmetry breaking bifurcation of solitons in an optical waveguide with focusing saturable nonlinearity and parity-time (𝒫𝒯)-symmetric complex-valued external potentials is investigated. As the soliton power increases, it is found that the branches of asymmetric solitons split off from the base branches of 𝒫𝒯-symmetric fundamental soliton. The bifurcation diagrams, consisting essentially of the propagation constants of optical solitons, indicate that symmetric fundamental and multipole solitons, as well as asymmetric solitons can exist. The stabilities and the dynamics characteristics of solitons are comprehensively investigated. We find the different instability scenarios of the symmetric solitons, but the symmetry breaking bifurcation is caused only by the onset of instability of the symmetric fundamental solitons. This result is further confirmed by the numerical examples with the different saturable nonlinearity parameters. In particular, we find that the soliton power and the stability of soliton at the bifurcation points are significantly changed by varying the strength of the saturable nonlinearities. These results provide additional way to control symmetry breaking bifurcations in 𝒫𝒯-symmetric optical waveguide.

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

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  1. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having 𝒫𝒯 symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  2. C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
    [Crossref]
  3. C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
    [Crossref]
  4. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32(17), 2632–2634 (2007).
    [Crossref] [PubMed]
  5. S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
    [Crossref] [PubMed]
  6. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).
  7. A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express,  21(18) 021651 (2013).
    [Crossref]
  8. H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89(7), 075136 (2014).
    [Crossref]
  9. Y. Choi, J. K. Hong, J. H. Cho, K. G. Lee, J. W. Yoon, and S. H. Song, “Parity-time-symmetry breaking in double-slab surface-plasmon-polariton waveguides,” Opt. Express,  23(9) 11783–11789 (2015).
    [Crossref] [PubMed]
  10. H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89(3), 033829 (2014).
    [Crossref]
  11. N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in 𝒫𝒯-symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110(5), 053901 (2013).
    [Crossref] [PubMed]
  12. N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
    [Crossref] [PubMed]
  13. 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(9), L171–L176 (2005).
    [Crossref]
  14. 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]
  15. 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]
  16. C. M. Huang, F. W. Ye, Y. V. Kartashov, B. A. Malomed, and X. F. Chen, “𝒫𝒯 symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
    [Crossref]
  17. 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]
  18. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805(R) (2011).
    [Crossref]
  19. X. Zhu, H. Wang, L. Zheng, H. Li, and Y. J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
    [Crossref] [PubMed]
  20. H. G. Li, Z. W. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
    [Crossref] [PubMed]
  21. S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
    [Crossref]
  22. B. Midya and R. Roychoudhury, “Nonlinear localized modes in 𝒫𝒯-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87(4), 045803 (2013).
    [Crossref]
  23. S. Nixon, L. J. Ge, and J. K. Yang, “Stability analysis for solitons in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
    [Crossref]
  24. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic 𝒫𝒯-symmetric potential,” Phys. Rev. A 85, 043840 (2012).
    [Crossref]
  25. V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
    [Crossref]
  26. 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(3), 033801 (2012).
    [Crossref]
  27. P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).
  28. U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
    [Crossref]
  29. P. F. Li, L. Li, and B. A. Malomed, “Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Phys. Rev. E 89(6), 062926 (2014).
    [Crossref]
  30. X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express,  22(15), 18401–18411 (2014).
    [Crossref] [PubMed]
  31. C. Q. Dai and X. G. Wang, “Light bullet in parity-time symmetric potential,” Nonlinear Dyn. 77(4), 1133–1139 (2014).
    [Crossref]
  32. Y. Chen and Z. Y. Yan, “Multi-dimensional stable fundamental solitons and excitations in PT-symmetric harmonic-Gaussian potentials with unbounded gain-and-loss distributions,” Commun. Nonlinear. Sci. Numer. Simulat. 57, 34–46 (2018).
    [Crossref]
  33. J. K. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39(19), 5547–5550 (2014).
    [Crossref] [PubMed]
  34. J. K. Yang, “Can parity-time-symmetric potentials support families of non-parity-time-symmetric solitons,” Stud. Appl. Math. 132(4), 332–353 (2014).
    [Crossref]
  35. J. K. Yang, “Symmetry breaking of solitons in two-dimensional complex potentials,” Phys. Rev. E 91(2), 023201 (2015).
    [Crossref]
  36. S. M. Hu and W. Hu, “Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices,” Physica B. 429, 28–32 (2013).
    [Crossref]
  37. P. Cao, X. Zhu, Y. J. He, and H. G. Li, “Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity,” Opt. Commun,  316, 190–197 (2014).
    [Crossref]
  38. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express. 12(13), 2831–2837 (2004).
    [Crossref] [PubMed]
  39. M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double-channel potential,” Phys. Rev. A 75(6), 063621 (2007).
    [Crossref]
  40. A. Sacchetti, “Universal critical power for nonlinear Schrödinger equations with symmetric double well potential,” Phys. Rev. Lett. 103(19), 194101 (2009).
    [Crossref]
  41. J. K. Yang, “Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations,” Stud. Appl. Math. 129(2), 133–162 (2012).
    [Crossref]
  42. J. K. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
    [Crossref]
  43. Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
    [Crossref]
  44. J. W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
    [Crossref] [PubMed]

2018 (1)

Y. Chen and Z. Y. Yan, “Multi-dimensional stable fundamental solitons and excitations in PT-symmetric harmonic-Gaussian potentials with unbounded gain-and-loss distributions,” Commun. Nonlinear. Sci. Numer. Simulat. 57, 34–46 (2018).
[Crossref]

2016 (1)

P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).

2015 (3)

J. K. Yang, “Symmetry breaking of solitons in two-dimensional complex potentials,” Phys. Rev. E 91(2), 023201 (2015).
[Crossref]

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

Y. Choi, J. K. Hong, J. H. Cho, K. G. Lee, J. W. Yoon, and S. H. Song, “Parity-time-symmetry breaking in double-slab surface-plasmon-polariton waveguides,” Opt. Express,  23(9) 11783–11789 (2015).
[Crossref] [PubMed]

2014 (9)

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89(3), 033829 (2014).
[Crossref]

C. M. Huang, F. W. Ye, Y. V. Kartashov, B. A. Malomed, and X. F. Chen, “𝒫𝒯 symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89(7), 075136 (2014).
[Crossref]

J. K. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39(19), 5547–5550 (2014).
[Crossref] [PubMed]

J. K. Yang, “Can parity-time-symmetric potentials support families of non-parity-time-symmetric solitons,” Stud. Appl. Math. 132(4), 332–353 (2014).
[Crossref]

P. F. Li, L. Li, and B. A. Malomed, “Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Phys. Rev. E 89(6), 062926 (2014).
[Crossref]

X. F. Zhu, Y. G. Peng, and D. G. Zhao, “Anisotropic reflection oscillation in periodic multilayer structures of parity-time symmetry,” Opt. Express,  22(15), 18401–18411 (2014).
[Crossref] [PubMed]

C. Q. Dai and X. G. Wang, “Light bullet in parity-time symmetric potential,” Nonlinear Dyn. 77(4), 1133–1139 (2014).
[Crossref]

P. Cao, X. Zhu, Y. J. He, and H. G. Li, “Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity,” Opt. Commun,  316, 190–197 (2014).
[Crossref]

2013 (5)

U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
[Crossref]

S. M. Hu and W. Hu, “Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices,” Physica B. 429, 28–32 (2013).
[Crossref]

B. Midya and R. Roychoudhury, “Nonlinear localized modes in 𝒫𝒯-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87(4), 045803 (2013).
[Crossref]

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express,  21(18) 021651 (2013).
[Crossref]

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 (6)

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
[Crossref] [PubMed]

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

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic 𝒫𝒯-symmetric potential,” Phys. Rev. A 85, 043840 (2012).
[Crossref]

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

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(3), 033801 (2012).
[Crossref]

J. K. Yang, “Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations,” Stud. Appl. Math. 129(2), 133–162 (2012).
[Crossref]

2011 (4)

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805(R) (2011).
[Crossref]

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

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

S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
[Crossref]

2010 (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]

2009 (2)

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]

A. Sacchetti, “Universal critical power for nonlinear Schrödinger equations with symmetric double well potential,” Phys. Rev. Lett. 103(19), 194101 (2009).
[Crossref]

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]

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

2007 (3)

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

M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double-channel potential,” Phys. Rev. A 75(6), 063621 (2007).
[Crossref]

J. K. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[Crossref]

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(9), L171–L176 (2005).
[Crossref]

2004 (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express. 12(13), 2831–2837 (2004).
[Crossref] [PubMed]

2003 (1)

J. W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[Crossref] [PubMed]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

1999 (1)

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

1998 (1)

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

1997 (1)

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
[Crossref]

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(4), 041805(R) (2011).
[Crossref]

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(3), 033801 (2012).
[Crossref]

Achilleos, V.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Aimez, V.

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

Alaeian, H.

H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89(7), 075136 (2014).
[Crossref]

H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89(3), 033829 (2014).
[Crossref]

Al-Marzoug, S. M.

U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
[Crossref]

Bagnato, V. S.

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

Bahlouli, H.

U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
[Crossref]

Baturina, T. I.

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
[Crossref] [PubMed]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

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

Benisty, H.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express,  21(18) 021651 (2013).
[Crossref]

Boettcher, S.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
[Crossref]

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

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Cao, P.

P. Cao, X. Zhu, Y. J. He, and H. G. Li, “Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity,” Opt. Commun,  316, 190–197 (2014).
[Crossref]

Carmon, T.

J. W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[Crossref] [PubMed]

Carretero-González, R.

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Chen, X. F.

Chen, Y.

Y. Chen and Z. Y. Yan, “Multi-dimensional stable fundamental solitons and excitations in PT-symmetric harmonic-Gaussian potentials with unbounded gain-and-loss distributions,” Commun. Nonlinear. Sci. Numer. Simulat. 57, 34–46 (2018).
[Crossref]

Chen, Z. G.

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
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Cho, J. H.

Choi, Y.

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(3), 033801 (2012).
[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]

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

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
[Crossref]

Chtchelkatchev, N. M.

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
[Crossref] [PubMed]

Coskun, T. H.

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
[Crossref]

Dai, C. Q.

C. Q. Dai and X. G. Wang, “Light bullet in parity-time symmetric potential,” Nonlinear Dyn. 77(4), 1133–1139 (2014).
[Crossref]

Degiron, A.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express,  21(18) 021651 (2013).
[Crossref]

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(9), L171–L176 (2005).
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H. Alaeian and J. A. Dionne, “Non-Hermitian nanophotonic and plasmonic waveguides,” Phys. Rev. B 89(7), 075136 (2014).
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H. Alaeian and J. A. Dionne, “Parity-time-symmetric plasmonic metamaterials,” Phys. Rev. A 89(3), 033829 (2014).
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Duchesne, D.

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

El-Ganainy, R.

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

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

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

Fleischer, J. W.

J. W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[Crossref] [PubMed]

Frantzeskakis, D. J.

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Ge, L. J.

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

Golubov, A. A.

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
[Crossref] [PubMed]

Gunther, U.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

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. J.

P. Cao, X. Zhu, Y. J. He, and H. G. Li, “Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity,” Opt. Commun,  316, 190–197 (2014).
[Crossref]

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

Hu, S. M.

S. M. Hu and W. Hu, “Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices,” Physica B. 429, 28–32 (2013).
[Crossref]

S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
[Crossref]

Hu, W.

S. M. Hu and W. Hu, “Defect solitons in saturable nonlinearity media with parity-time symmetric optical lattices,” Physica B. 429, 28–32 (2013).
[Crossref]

S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
[Crossref]

Huang, C. M.

Jiang, X.

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
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Kartashov, Y. V.

C. M. Huang, F. W. Ye, Y. V. Kartashov, B. A. Malomed, and X. F. Chen, “𝒫𝒯 symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805(R) (2011).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express. 12(13), 2831–2837 (2004).
[Crossref] [PubMed]

Kevrekidis, P. G.

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in 𝒫𝒯-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear 𝒫𝒯 phase transitions,” Phys. Rev. A 86(1), 013808 (2012).
[Crossref]

Khawaja, U. A.

U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
[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. S.

U. A. Khawaja, S. M. Al-Marzoug, H. Bahlouli, and Y. S. Kivshar, “Unidirectional soliton flows in 𝒫𝒯-symmetric potentials,” Phys. Rev. A 88(2), 023830 (2013).
[Crossref]

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
[Crossref]

Klaiman, S.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic 𝒫𝒯-symmetric potential,” Phys. Rev. A 85, 043840 (2012).
[Crossref]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯-symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805(R) (2011).
[Crossref]

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(3), 033801 (2012).
[Crossref]

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(3), 033801 (2012).
[Crossref]

Lakoba, T. I.

J. K. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
[Crossref]

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]

Lee, K. G.

Li, H.

Li, H. G.

P. Cao, X. Zhu, Y. J. He, and H. G. Li, “Gap solitons supported by parity-time-symmetric optical lattices with defocusing saturable nonlinearity,” Opt. Commun,  316, 190–197 (2014).
[Crossref]

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

Li, L.

P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).

P. F. Li, L. Li, and B. A. Malomed, “Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Phys. Rev. E 89(6), 062926 (2014).
[Crossref]

Li, P. F.

P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).

P. F. Li, L. Li, and B. A. Malomed, “Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Phys. Rev. E 89(6), 062926 (2014).
[Crossref]

Liu, B.

P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).

Lu, D.

S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
[Crossref]

Lupu, A.

A. Lupu, H. Benisty, and A. Degiron, “Switching using PT symmetry in plasmonic systems: positive role of the losses,” Opt. Express,  21(18) 021651 (2013).
[Crossref]

Ma, X.

S. M. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex 𝒫𝒯-symmetric Gaussian potentials,” Phys. Rev. A,  84(4), 043818 (2011).
[Crossref]

Makris, K. G.

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

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

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

Malomed, B. A.

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

P. F. Li, L. Li, and B. A. Malomed, “Multisoliton Newton’s cradles and supersolitons in regular and parity-time-symmetric nonlinear couplers,” Phys. Rev. E 89(6), 062926 (2014).
[Crossref]

C. M. Huang, F. W. Ye, Y. V. Kartashov, B. A. Malomed, and X. F. Chen, “𝒫𝒯 symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39(18), 5443–5446 (2014).
[Crossref]

M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double-channel potential,” Phys. Rev. A 75(6), 063621 (2007).
[Crossref]

Matuszewski, M.

M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double-channel potential,” Phys. Rev. A 75(6), 063621 (2007).
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Meisinger, P. N.

C. M. Bender, S. Boettcher, and P. N. Meisinger, “PT-symmetric quantum mechanics,” J. Math. Phys. 40(5), 2201–2229 (1999).
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Midya, B.

B. Midya and R. Roychoudhury, “Nonlinear localized modes in 𝒫𝒯-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87(4), 045803 (2013).
[Crossref]

Mihalache, D.

P. F. Li, B. Liu, L. Li, and D. Mihalache, “Nonlinear parity-time-symmetry breaking in optical waveguides with complex Gaussian-type potentials,” Rom. J. Phys. 61(3–4), 577–594 (2016).

V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, “Bose-Einstein condensation: twenty years after,” Rom. Rep. Phys. 67(1), 5–50 (2015).

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(3), 033801 (2012).
[Crossref]

Moiseyev, N.

S. Klaiman, U. Gunther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯-symmetric waveguides,” Phys. Rev. Lett. 101(8), 080402 (2008).
[Crossref] [PubMed]

Morandotti, R.

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

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(9), L171–L176 (2005).
[Crossref]

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]

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

Nixon, S.

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

Peng, Y. G.

Roychoudhury, R.

B. Midya and R. Roychoudhury, “Nonlinear localized modes in 𝒫𝒯-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87(4), 045803 (2013).
[Crossref]

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(9), L171–L176 (2005).
[Crossref]

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]

Sacchetti, A.

A. Sacchetti, “Universal critical power for nonlinear Schrödinger equations with symmetric double well potential,” Phys. Rev. Lett. 103(19), 194101 (2009).
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Salamo, G. J.

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

Segev, M.

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

J. W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003).
[Crossref] [PubMed]

Z. G. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, and Y. S. Kivshar, “Coupled photorefractive spatial-soliton pairs,” J. Opt. Soc. Am. B. 14(11), 3066–3077 (1997).
[Crossref]

Shi, Z. W.

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]

Song, S. H.

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express. 12(13), 2831–2837 (2004).
[Crossref] [PubMed]

Trippenbach, M.

M. Matuszewski, B. A. Malomed, and M. Trippenbach, “Spontaneous symmetry breaking of solitons trapped in a double-channel potential,” Phys. Rev. A 75(6), 063621 (2007).
[Crossref]

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]

Vinokur, V. M.

N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V. M. Vinokur, “Stimulation of the fluctuation superconductivity by 𝒫𝒯 symmetry,” Phys. Rev. Lett. 109(15), 150405 (2012).
[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.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express. 12(13), 2831–2837 (2004).
[Crossref] [PubMed]

Wang, H.

Wang, X. G.

C. Q. Dai and X. G. Wang, “Light bullet in parity-time symmetric potential,” Nonlinear Dyn. 77(4), 1133–1139 (2014).
[Crossref]

Yan, Z. Y.

Y. Chen and Z. Y. Yan, “Multi-dimensional stable fundamental solitons and excitations in PT-symmetric harmonic-Gaussian potentials with unbounded gain-and-loss distributions,” Commun. Nonlinear. Sci. Numer. Simulat. 57, 34–46 (2018).
[Crossref]

Yang, J. K.

J. K. Yang, “Symmetry breaking of solitons in two-dimensional complex potentials,” Phys. Rev. E 91(2), 023201 (2015).
[Crossref]

J. K. Yang, “Can parity-time-symmetric potentials support families of non-parity-time-symmetric solitons,” Stud. Appl. Math. 132(4), 332–353 (2014).
[Crossref]

J. K. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39(19), 5547–5550 (2014).
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J. K. Yang, “Classification of solitary wave bifurcations in generalized nonlinear Schrödinger equations,” Stud. Appl. Math. 129(2), 133–162 (2012).
[Crossref]

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

J. K. Yang and T. I. Lakoba, “Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations,” Stud. Appl. Math. 118(2), 153–197 (2007).
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Figures (7)

Fig. 1
Fig. 1 Diagram of bifurcation and the corresponding symmetric and asymmetric solitons. (a) Profiles of the real and even function g (ξ) (black dashed curve), real (blue solid curve) and imaginary (thin red solid curve) components of the potential. (b) The propagation constants versus the soliton power, where the symmetric fundamental solitons (blue solid curve labeled with SS1), symmetric dipole solitons for the excited state (black dash-dot curve labeled with SS2), symmetric tripole solitons for the excited state (thin gray solid curve labeled with SS3), as well as asymmetric solitons (red dotted curve labeled with AS) are shown. (c) and (d) Distributions of fundamental symmetric and asymmetric solitons with the soliton power P = 1.5, where the blue solid and dashed red curves represent the real part and imaginary part, respectively. Here, the parameters are ξ0 = 2, χ0 = 1, and W0 = 1.5.
Fig. 2
Fig. 2 Profiles of solitons for different soliton power. (a) Symmetric fundamental solitons, (b) symmetric dipole solitons for the excited state, (c) symmetric tripole solitons, (d) asymmetric solitons. Here, the parameters are the same as those in Fig. 1.
Fig. 3
Fig. 3 Diagrams of bifurcation and profiles of solitons for different saturable nonlinear parameters. (a), (b) and (c) are diagrams of bifurcations with saturable nonlinear parameters S = 0.1, S = 0.5, and S = 1, respectively, where the vertical dashed lines show the soliton powers at bifurcation points. (d), (e) and (f) depict that the profiles transform from the symmetric fundamental solitons into asymmetric solitons due to symmetry breaking. The vertical plane shows the soliton power at bifurcation points, where the saturable nonlinear parameters are S = 0.1, S = 0.5, and S = 1, respectively. Here, the other parameters are the same as those in Fig. 1.
Fig. 4
Fig. 4 Diagrams of bifurcation depending on the different modulation strength of the potential W0 with different saturable nonlinear parameters. (a), (b) and (c) depict the diagrams of bifurcation with modulation strength of the potential W0 = 2, and the focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. The red dotted curves and blue solid curves represent the propagation constants of symmetric and asymmetric solitons. (d), (e) and (f) show the diagrams of bifurcation with modulation strength of the potential W0 = 1, the focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. Here, the other parameters of potential are the same as those in Fig. 1.
Fig. 5
Fig. 5 Eigenvalues of linear stability analysis for the different parameters of saturable nonlinearity. (a), (b) and (c) depict the largest real part of δ for the symmetric and asymmetric solitons with focusing saturable nonlinearity parameters S = 0.1, 0.5 and 1, respectively. Here, the parameters are the same as in Fig. 3.
Fig. 6
Fig. 6 The eigenvalue spectra of linear stability analysis for the solitons with saturable parameter S = 1 and the corresponding evolution plots. (a), (b) and (c) show the eigenvalue spectra of linear stability analysis for SS1, SS2 and SS3, (d) shows the eigenvalue spectrum of AS with the soliton power P = 1, respectively. (e), (f), (g), and (h) The corresponding evolution plots of the field intensity. Here the other parameters are the same as in Fig. 1.
Fig. 7
Fig. 7 The eigenvalue spectra of linear stability analysis for the solitons with saturable parameter S = 1 and the corresponding evolution plots. (a), (b) and (c) show the eigenvalue spectra of linear stability analysis for SS1, SS2 and SS3, (d) shows the eigenvalue spectrum of AS with the soliton power P = 2.5. (e), (f), (g), and (h) The corresponding evolution plots of the field intensity. Here the other parameters are the same as those in Fig. 1.

Equations (12)

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i ψ ζ + 2 ψ ξ 2 + U ( ξ ) ψ + σ | ψ | 2 ψ 1 + S | ψ | 2 = 0 ,
d 2 ϕ d ξ 2 + U ( ξ ) ϕ ( ξ ) + σ | ϕ | 2 ϕ 1 + S | ϕ | 2 β ϕ ( ξ ) = 0 .
U ( ξ ) = g 2 ( ξ ) + i d g ( ξ ) d ξ ,
g ( ξ ) = W 0 [ sech ( ξ + ξ 0 χ 0 ) + sech ( ξ ξ 0 χ 0 ) ] ,
V ( ξ ) = W 0 2 [ sech ( ξ + ξ 0 χ 0 ) + sech ( ξ ξ 0 χ 0 ) ] 2 ,
W ( ξ ) = W 0 χ 0 [ sech ( ξ + ξ 0 χ 0 ) tanh ( ξ + ξ 0 χ 0 ) + sech ( ξ ξ 0 χ 0 ) tanh ( ξ ξ 0 χ 0 ) ] .
ψ ( ζ , ξ ) = e i β ζ [ ϕ ( ξ ) + u ( ξ ) e δ ζ + v * ( ξ ) e δ * ζ ] ,
i ( L 11 L 12 L 21 L 22 ) ( u v ) = δ ( u v ) .
L 11 = d 2 d ξ 2 + U β + 2 σ | ϕ | 2 1 + S | ϕ | 2 σ S | ϕ | 4 ( 1 + S | ϕ | 2 ) 2 ,
L 12 = 2 σ ϕ 2 1 + S | ϕ | 2 σ S | ϕ | 2 ϕ 2 ( 1 + S | ϕ | 2 ) 2 ,
L 21 = σ S | ϕ | 2 ϕ * 2 ( 1 + S | ϕ | 2 ) 2 2 σ ϕ * 2 1 + S | ϕ | 2 ,
L 22 = d 2 d ξ 2 U * + β 2 σ | ϕ | 2 1 + S | ϕ | 2 + σ S | ϕ | 4 ( 1 + S | ϕ | 2 ) 2 ,

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