Abstract

We theoretically report the influence of a class of near-parity-time-(𝒫𝒯-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter α2 or nonlinear gain-loss coefficient β2, we do find stable exact solitons in the second quadrant of the (α2, β2) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.

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

Full Article  |  PDF Article
OSA Recommended Articles
Impact of phase on collision between vortex solitons in three-dimensional cubic-quintic complex Ginzburg-Landau equation

Bin Liu, Yun-Feng Liu, and Xing-Dao He
Opt. Express 22(21) 26203-26211 (2014)

One- and two-dimensional modes in the complex Ginzburg-Landau equation with a trapping potential

Thawatchai Mayteevarunyoo, Boris A. Malomed, and Dmitry V. Skryabin
Opt. Express 26(7) 8849-8865 (2018)

References

  • View by:
  • |
  • |
  • |

  1. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
    [Crossref]
  2. M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
    [Crossref]
  3. M. van Hecke, “Coherent and incoherent structures in systems described by the 1d cgle: Experiments and identification,” Phys. D 174, 134–151 (2003).
    [Crossref]
  4. M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
    [Crossref]
  5. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004).
    [Crossref]
  6. N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
    [Crossref]
  7. C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007).
    [Crossref]
  8. N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
    [Crossref] [PubMed]
  9. Y. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
    [Crossref]
  10. D. Mihalache, “Localized structures in nonlinear optical media: a selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).
  11. N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
    [Crossref]
  12. N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003).
    [Crossref]
  13. J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005).
    [Crossref]
  14. E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005).
    [Crossref]
  15. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
    [Crossref]
  16. V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
    [Crossref]
  17. D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
    [Crossref] [PubMed]
  18. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
    [Crossref]
  19. Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013).
    [Crossref]
  20. 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).
    [Crossref]
  21. Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
    [Crossref] [PubMed]
  22. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998).
    [Crossref]
  23. C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
    [Crossref]
  24. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007).
    [Crossref]
  25. Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
    [Crossref]
  26. Z. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [Crossref] [PubMed]
  27. Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
    [Crossref]
  28. Z. Yan, “Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation,” Philos. Trans. R. Soc. London, Ser. A 371, 20120059 (2013).
    [Crossref]
  29. Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
    [Crossref]
  30. Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
    [Crossref]
  31. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
    [Crossref]
  32. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
    [Crossref]
  33. V. Achilleos, P. Kevrekidis, D. 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).
    [Crossref]
  34. 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).
    [Crossref]
  35. A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
    [Crossref] [PubMed]
  36. 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, 192–195 (2010).
    [Crossref]
  37. 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).
    [Crossref] [PubMed]
  38. G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
    [Crossref] [PubMed]
  39. A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
    [Crossref] [PubMed]
  40. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
    [Crossref]
  41. A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
    [Crossref]
  42. P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016).
    [Crossref]
  43. K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017).
    [Crossref]
  44. E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice solitons,” Opt. Lett. 29, 845–847 (2004).
    [Crossref] [PubMed]
  45. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [Crossref] [PubMed]
  46. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
    [Crossref]
  47. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [Crossref]
  48. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840 (2012).
    [Crossref]
  49. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
    [Crossref]
  50. C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
    [Crossref]
  51. B. Midya and R. Roychoudhury, “Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87, 045803 (2013).
    [Crossref]
  52. S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011).
    [Crossref]
  53. J. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39, 5547–5550 (2014).
    [Crossref] [PubMed]
  54. C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
    [Crossref]
  55. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011).
    [Crossref]
  56. N. Moiseyev, “Crossing rule for a PT-symmetric two-level time-periodic system,” Phys. Rev. A 83, 052125 (2011).
    [Crossref]
  57. 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).
    [Crossref]
  58. H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016).
    [Crossref]
  59. S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
    [Crossref]
  60. H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012).
    [Crossref]
  61. F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
    [Crossref]
  62. 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).
    [Crossref]
  63. Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
    [Crossref]
  64. R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
    [Crossref]
  65. D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
    [Crossref]
  66. C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
    [Crossref]
  67. Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
    [Crossref]
  68. Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016).
    [Crossref]
  69. Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
    [Crossref]
  70. Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017).
    [Crossref]
  71. Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017).
    [Crossref]
  72. Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017).
    [Crossref]
  73. Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
    [Crossref]
  74. J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
    [Crossref]
  75. E. N. Tsoy, I. M. Allayarov, and F. K. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39, 4215–4218 (2014).
    [Crossref] [PubMed]
  76. V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39, 5535–5538 (2014).
    [Crossref] [PubMed]
  77. S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136, 459–483 (2016).
    [Crossref]
  78. Y. Kominis, “Soliton dynamics in symmetric and non-symmetric complex potentials,” Opt. Commun. 334, 265–272 (2015).
    [Crossref]
  79. Y. Kominis, “Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes,” Phys. Rev. A 92, 063849 (2015).
    [Crossref]
  80. J. Yang and S. Nixon, “Stability of soliton families in nonlinear schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380, 3803–3809 (2016).
    [Crossref]
  81. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016).
    [Crossref]
  82. J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
    [Crossref]
  83. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
    [Crossref]
  84. N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995).
    [Crossref] [PubMed]
  85. N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
    [Crossref]
  86. J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
    [Crossref]

2018 (2)

Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
[Crossref]

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

2017 (6)

K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017).
[Crossref]

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017).
[Crossref]

Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017).
[Crossref]

Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017).
[Crossref]

2016 (8)

Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
[Crossref]

Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016).
[Crossref]

H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016).
[Crossref]

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016).
[Crossref]

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

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

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

2015 (7)

Y. Kominis, “Soliton dynamics in symmetric and non-symmetric complex potentials,” Opt. Commun. 334, 265–272 (2015).
[Crossref]

Y. Kominis, “Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes,” Phys. Rev. A 92, 063849 (2015).
[Crossref]

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

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

Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
[Crossref]

Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
[Crossref]

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
[Crossref]

2014 (11)

Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
[Crossref] [PubMed]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

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).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

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

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
[Crossref]

E. N. Tsoy, I. M. Allayarov, and F. K. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39, 4215–4218 (2014).
[Crossref] [PubMed]

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

2013 (9)

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

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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

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).
[Crossref]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[Crossref]

Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013).
[Crossref]

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).
[Crossref]

Z. Yan, “Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation,” Philos. Trans. R. Soc. London, Ser. A 371, 20120059 (2013).
[Crossref]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

2012 (6)

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

V. Achilleos, P. Kevrekidis, D. 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).
[Crossref]

Y. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[Crossref]

H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012).
[Crossref]

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).
[Crossref] [PubMed]

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

2011 (5)

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

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

N. Moiseyev, “Crossing rule for a PT-symmetric two-level time-periodic system,” Phys. Rev. A 83, 052125 (2011).
[Crossref]

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

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

2010 (3)

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, 192–195 (2010).
[Crossref]

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

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

2009 (1)

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

2008 (3)

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

2007 (4)

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

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

C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007).
[Crossref]

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[Crossref] [PubMed]

2005 (4)

N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005).
[Crossref]

E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005).
[Crossref]

2004 (2)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004).
[Crossref]

E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice solitons,” Opt. Lett. 29, 845–847 (2004).
[Crossref] [PubMed]

2003 (3)

M. van Hecke, “Coherent and incoherent structures in systems described by the 1d cgle: Experiments and identification,” Phys. D 174, 134–151 (2003).
[Crossref]

N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
[Crossref]

2002 (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

2001 (3)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
[Crossref]

2000 (2)

M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
[Crossref]

M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
[Crossref]

1998 (1)

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

1997 (1)

N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
[Crossref]

1996 (1)

N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
[Crossref]

1995 (1)

N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995).
[Crossref] [PubMed]

Abdullaev, F. K.

E. N. Tsoy, I. M. Allayarov, and F. K. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39, 4215–4218 (2014).
[Crossref] [PubMed]

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

Achilleos, V.

V. Achilleos, P. Kevrekidis, D. 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).
[Crossref]

Afanasjev, V.

N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
[Crossref]

N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995).
[Crossref] [PubMed]

Ahmed, Z.

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[Crossref]

Aimez, V.

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

Akhmediev, N.

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[Crossref] [PubMed]

E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005).
[Crossref]

J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005).
[Crossref]

N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
[Crossref]

N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
[Crossref]

N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
[Crossref]

N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995).
[Crossref] [PubMed]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[Crossref]

Alberucci, A.

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
[Crossref]

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).
[Crossref]

Aleksic, N.

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

Allayarov, I. M.

Alù, A.

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

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
[Crossref]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[Crossref]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

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).
[Crossref]

Bender, C. M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

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

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
[Crossref]

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

Bersch, C.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Bludov, Y. V.

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[Crossref]

Boettcher, S.

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

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).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
[Crossref]

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).
[Crossref]

Carretero-González, R.

V. Achilleos, P. Kevrekidis, D. 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).
[Crossref]

Cartarius, H.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012).
[Crossref]

Castaldi, G.

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

Chen, L.

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

Chen, P.-Y.

P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016).
[Crossref]

Chen, Y.

Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
[Crossref]

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017).
[Crossref]

Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017).
[Crossref]

Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
[Crossref]

Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016).
[Crossref]

Chiang, K. S.

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
[Crossref]

Christodoulides, D.

H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016).
[Crossref]

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

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

Christodoulides, D. N.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

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

E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice solitons,” Opt. Lett. 29, 845–847 (2004).
[Crossref] [PubMed]

Crasovan, L.-C.

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Dai, C.-Q.

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Dast, D.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

Devassy, L.

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
[Crossref]

Dizdarevic, D.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

Dmitriev, S. V.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Dorofeenko, A. V.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Duchesne, D.

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

El-Ganainy, R.

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

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

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

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

Engheta, N.

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

Facao, M. M.

M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
[Crossref]

Fan, S.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Fan, Y.

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

Fedorov, S.

N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
[Crossref]

Ferreira, M. F.

M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
[Crossref]

Fortanier, R.

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

Frantzeskakis, D.

V. Achilleos, P. Kevrekidis, D. 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).
[Crossref]

Galdi, V.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 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, 023822 (2012).
[Crossref]

Gianfreda, M.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Grelu, P.

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[Crossref] [PubMed]

Guo, A.

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

Gutöhrlein, R.

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

Haag, D.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

Hang, C.

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
[Crossref]

He, Y.

Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
[Crossref] [PubMed]

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).
[Crossref]

Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013).
[Crossref]

Y. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[Crossref]

Hu, S.

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

Hu, W.

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

Huang, J.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Ipsen, M.

M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
[Crossref]

Jiang, X.

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

Jisha, C. P.

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
[Crossref]

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).
[Crossref]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
[Crossref]

Jung, J.

P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016).
[Crossref]

Kartashov, Y. V.

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

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Kevrekidis, P.

V. Achilleos, P. Kevrekidis, D. 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).
[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, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Kominis, Y.

Y. Kominis, “Soliton dynamics in symmetric and non-symmetric complex potentials,” Opt. Commun. 334, 265–272 (2015).
[Crossref]

Y. Kominis, “Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes,” Phys. Rev. A 92, 063849 (2015).
[Crossref]

Konotop, V. V.

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

Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
[Crossref]

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

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[Crossref]

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, 041805 (2011).
[Crossref]

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
[Crossref]

Kuriakose, V.

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014).
[Crossref]

Larionova, Y.

C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007).
[Crossref]

Latas, S. C.

M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
[Crossref]

Leblond, H.

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Lee, C.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Lei, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Li, H.

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).
[Crossref]

Li, X.

Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
[Crossref]

Lisyansky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Long, G. L.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Lu, D.

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

Lumer, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Ma, X.

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

Main, J.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

Makris, K.

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

Makris, K. G.

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

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Malomed, B.

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Malomed, B. A.

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
[Crossref] [PubMed]

Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013).
[Crossref]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[Crossref]

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).
[Crossref]

Mandel, P.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004).
[Crossref]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Midya, B.

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

Mihalache, D.

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

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

Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
[Crossref] [PubMed]

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).
[Crossref]

Y. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012).
[Crossref]

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Miri, M.-A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Moiseyev, N.

N. Moiseyev, “Crossing rule for a PT-symmetric two-level time-periodic system,” Phys. Rev. A 83, 052125 (2011).
[Crossref]

Monifi, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Morandotti, R.

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

Musslimani, Z.

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

Musslimani, Z. H.

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Näger, J.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

Nixon, S.

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

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

Nixon, S. D.

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

Nori, F.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Notomi, M.

K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017).
[Crossref]

Onishchukov, G.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Özdemir, S. K.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Peng, B.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Peschel, U.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Plotnik, Y.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Pukhov, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Rechtsman, M. C.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

Regensburger, A.

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

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).
[Crossref] [PubMed]

Rosanov, N.

N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
[Crossref]

Roychoudhury, R.

B. Midya and R. Roychoudhury, “Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87, 045803 (2013).
[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, 192–195 (2010).
[Crossref]

Salamo, G.

A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 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, 173901 (2013).
[Crossref] [PubMed]

Segev, M.

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[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, 192–195 (2010).
[Crossref]

Shatsev, A.

N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
[Crossref]

Shen, J.

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

Shi, Z.

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

Single, F.

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

Siviloglou, G.

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

Skarka, V.

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

Sørensen, P. G.

M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
[Crossref]

Soto-Crespo, J.

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[Crossref] [PubMed]

N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
[Crossref]

N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
[Crossref]

Soto-Crespo, J. M.

N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
[Crossref]

Soto-Crespo, J.-M.

J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005).
[Crossref]

Stegeman, G. I.

Suchkov, S. V.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Sukhorukov, A. A.

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Takata, K.

K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017).
[Crossref]

Tlidi, M.

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004).
[Crossref]

Torner, L.

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

Tsoy, E. N.

E. N. Tsoy, I. M. Allayarov, and F. K. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39, 4215–4218 (2014).
[Crossref] [PubMed]

E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005).
[Crossref]

Ultanir, E. A.

van Hecke, M.

M. van Hecke, “Coherent and incoherent structures in systems described by the 1d cgle: Experiments and identification,” Phys. D 174, 134–151 (2003).
[Crossref]

Vinogradov, A. P.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Volatier-Ravat, M.

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

Wang, H.

H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016).
[Crossref]

Wang, X.-G.

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Weiss, C.

C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007).
[Crossref]

Wen, Z.

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017).
[Crossref]

Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
[Crossref]

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
[Crossref]

Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
[Crossref]

Wen, Z.-C.

Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
[Crossref]

Wunner, G.

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012).
[Crossref]

Yan, Z.

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
[Crossref]

Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017).
[Crossref]

Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017).
[Crossref]

Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017).
[Crossref]

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016).
[Crossref]

Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
[Crossref]

Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
[Crossref]

Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
[Crossref]

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
[Crossref]

Z. Yan, “Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation,” Philos. Trans. R. Soc. London, Ser. A 371, 20120059 (2013).
[Crossref]

Yang, J.

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

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

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

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

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

J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
[Crossref]

Yang, L.

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Yang, Z.

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

Zezyulin, D. A.

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

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

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, 041805 (2011).
[Crossref]

Zhang, X.-F.

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

Zheng, Y.

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

Zhou, G.-Q.

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

Zhu, X.

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

Zyablovsky, A. A.

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Am. J. Phys (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003).
[Crossref]

Appl. Phys. B (1)

N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005).
[Crossref]

Chaos (5)

N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007).
[Crossref] [PubMed]

Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016).
[Crossref]

Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017).
[Crossref]

Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017).
[Crossref]

J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (3)

Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018).
[Crossref]

C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017).
[Crossref]

H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016).
[Crossref]

Fiber & Integr. Opt. (1)

M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000).
[Crossref]

Int. J. Theor. Phys. (1)

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

J. Opt. B (1)

P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004).
[Crossref]

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008).
[Crossref]

Laser Photonics Rev. (1)

S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016).
[Crossref]

Math. Comput. Simul (1)

J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005).
[Crossref]

Nat. Phys. (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, 192–195 (2010).
[Crossref]

B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]

Nature (1)

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

Opt. Commun. (1)

Y. Kominis, “Soliton dynamics in symmetric and non-symmetric complex potentials,” Opt. Commun. 334, 265–272 (2015).
[Crossref]

Opt. Lett. (4)

Phil. Trans. R. Soc. A (1)

Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014).
[Crossref] [PubMed]

Philos. Trans. R. Soc. London, Ser. A (1)

Z. Yan, “Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation,” Philos. Trans. R. Soc. London, Ser. A 371, 20120059 (2013).
[Crossref]

Phys. D (1)

M. van Hecke, “Coherent and incoherent structures in systems described by the 1d cgle: Experiments and identification,” Phys. D 174, 134–151 (2003).
[Crossref]

Phys. Lett. A (6)

E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005).
[Crossref]

Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015).
[Crossref]

N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003).
[Crossref]

Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001).
[Crossref]

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

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001).
[Crossref]

Phys. Rep. (1)

M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000).
[Crossref]

Phys. Rev. A (21)

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007).
[Crossref]

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).
[Crossref]

Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015).
[Crossref]

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

V. Achilleos, P. Kevrekidis, D. 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).
[Crossref]

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

C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (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, 041805 (2011).
[Crossref]

N. Moiseyev, “Crossing rule for a PT-symmetric two-level time-periodic system,” Phys. Rev. A 83, 052125 (2011).
[Crossref]

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).
[Crossref]

H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012).
[Crossref]

F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014).
[Crossref]

Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013).
[Crossref]

R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014).
[Crossref]

D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015).
[Crossref]

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

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

C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014).
[Crossref]

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

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

Y. Kominis, “Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes,” Phys. Rev. A 92, 063849 (2015).
[Crossref]

Phys. Rev. Appl (2)

P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016).
[Crossref]

K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017).
[Crossref]

Phys. Rev. E (6)

Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015).
[Crossref]

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).
[Crossref]

Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017).
[Crossref]

N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996).
[Crossref]

Phys. Rev. Lett. (11)

N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995).
[Crossref] [PubMed]

V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010).
[Crossref]

D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005).
[Crossref] [PubMed]

N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997).
[Crossref]

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

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
[Crossref]

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

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

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

A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref] [PubMed]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref] [PubMed]

Phys. Usp. (1)

A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014).
[Crossref]

Rep. Prog. Phys. (2)

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

C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007).
[Crossref]

Rev. Mod. Phys. (2)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002).
[Crossref]

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

Rom. Rep. Phys. (1)

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

Sci. Rep. (2)

Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016).
[Crossref]

Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017).
[Crossref]

Stud. Appl. Math. (1)

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

Other (2)

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[Crossref]

J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010).
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Schematic of an experimental design apparatus described by Eq. (2).
Fig. 2
Fig. 2 Real and imaginary components of the first two lowest energy eigenvalues λ of the linear spectral problem (7) as a function of W0 at V0 = 1. (a1, a2) (α2, β2) = (0, 0), (b1, b2) (α2, β2) = (0, 0.1), (c1, c2) (α2, β2) = (−0.1, 0), (d1, d2) (α2, β2) = (−0.1, 0.1), in the near ����-symmetric potential (6).
Fig. 3
Fig. 3 Profiles of the near ����-symmetric potential (6) and soliton solutions with α2 = −1, β2 = 1. (a, b) W0 = 0.1, (c, d) W0 = 1.5. Evolutions of the exact solitons (8) with W0 = 0.1 in the second row while W0 = 1.5 in the third row: (a1, b1) (α2, β2) = (0, 0), (a2, b2) (α2, β2) = (0, 1), (a3, b3) (α2, β2) = (−1, 0), (a4, b4) (α2, β2) = (−1, 1), (a5, b5) (α2, β2) = (−0.2, −0.01). Unstable evolutions with W0 = 0.1 in the last row: (c1) (α2, β2) = (1, 1), (c2) (α2, β2) = (1, 0), (c3) (α2, β2) = (1, −1), (c4) (α2, β2) = (0, −1), (c5) (α2, β2) = (−1, −1).
Fig. 4
Fig. 4 Linear-stability maps [cf. Eq. (5)] of the exact solitons (8) in the (α2, β2) space [only black and dark regions denote stable solitons]: (a) W0 = 0, (b) W0 = 0.1, (c, d) W0 = 1.5, where (d) clearly indicates the concrete linear-stability situation around the original point in (c). Linear-stability spectra with W0 = 0.1: (a1) (α2, β2) = (0, 1), (a2) (α2, β2) = (0, −1), (b1) (α2, β2) = (−1, 0), (b2) (α2, β2) = (1, 0), (c1) (α2, β2) = (−1, 1), (c2) (α2, β2) = (1, −1), (d1) (α2, β2) = (−1, −1), (d2) (α2, β2) = (1, 1).
Fig. 5
Fig. 5 Collisions between the bright soliton (8) and boosted sech-shaped or rational solitary pulse, produced by the simulation of Eq. (2), with the initial input A(x, 0) = ϕ(x) + sech(x + 20) e4ix. (a) (α2, β2) = (0, 0), (b) (α2, β2) = (0, 1), (c) (α2, β2) = (−0.01, 0), (d) (α2, β2) = (−0.01, 1). Here ϕ(x) is given by Eq. (8) with V0 = 1,W0 = 0.1.
Fig. 6
Fig. 6 The density of energy generation E and energy flux j: (a, b) The same parameters are used as Figs. 3(a) and 3(b); (c, d) the same parameters as Figs. 3(c) and 3(d). Here ‘G’ (‘L’) denotes the gain (loss) region.
Fig. 7
Fig. 7 Excitations of exact nonlinear localized modes [cf. Eq. (10)]. (a) α21 = β2 = 0, α22 = −1, (b) α2 = β21 = 0, β22 = 1, (c) α21 = β21 = 0, α22 = −1, β22 = 1, other parameter is W0 = 1.5; (d) W01 = 0.1, W02 = 1.5, other parameters are α2 = −0.2, β2 = −0.01.

Equations (11)

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

i A z + ( α 1 + i α 2 ) A x x + i γ A + ( β 1 + i β 2 ) | A | 2 A = 0 ,
i A z + ( α 1 + i α 2 ) A x x + [ V ( x ) + i W ( x ) ] A + ( β 1 + i β 2 ) | A | 2 A = 0 ,
[ ( α 1 + i α 2 ) d 2 d x 2 + V ( x ) + i W ( x ) + ( β 1 + i β 2 ) | ϕ | 2 ] ϕ = q ϕ ,
A ( x , z ) = { ϕ ( x ) + [ f ( x ) e δ z + g * ( x ) e δ * z ] } e i q z ,
i [ L ^ 1 L ^ 2 L ^ 2 * L ^ 1 * ] [ f ( x ) g ( x ) ] = δ [ f ( x ) g ( x ) ] ,
V ( x ) = V 0 sech 2 ( x ) α 2 α 1 W 0 sech ( x ) tanh ( x ) , W ( x ) = W 0 sech ( x ) tanh ( x ) + W 1 sech 2 ( x ) α 2 ,
L Φ ( x ) = λ Φ ( x ) , L = ( α 1 + i α 2 ) x 2 + V ( x ) + i W ( x ) ,
ϕ ( x ) = ( α 1 [ 2 + W 0 2 / ( 9 α 1 2 ) ] V 0 ) / β 1 sech ( x ) exp { i W 0 3 α 1 tan 1 [ sinh ( x ) ] } .
E = 2 α 2 | A x | 2 α 2 ( | A | 2 ) x x 2 W ( x ) | A | 2 2 β 2 | A | 4 ,
i A 2 + [ α 1 + i α 2 ( z ) ] A x x + [ V ( x , z ) ) + i W ( x , z ) ] A + [ β 1 + i β 2 ( z ) ] | A | 2 A = 0 ,
( z ) = { 1 2 ( 2 1 ) [ 1 cos ( π z / 1000 ) ] + 1 , 0 z < 1000 , 2 , z 1000

Metrics