Abstract

Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded PT -symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.

© 2014 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
    [CrossRef]
  2. D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
    [CrossRef]
  3. I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
    [CrossRef]
  4. Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
    [CrossRef]
  5. D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
    [CrossRef] [PubMed]
  6. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  7. J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
    [CrossRef]
  8. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
    [CrossRef]
  9. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
    [CrossRef]
  10. A. Szameit, S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010).
    [CrossRef]
  11. T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
    [CrossRef]
  12. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef] [PubMed]
  13. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
    [CrossRef] [PubMed]
  14. B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
    [CrossRef]
  15. P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
    [CrossRef]
  16. P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
    [CrossRef]
  17. M. Öster, M. Johansson, “Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2,” Phys. Rev. E 73, 066608 (2006).
    [CrossRef]
  18. C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
    [CrossRef]
  19. D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
    [CrossRef] [PubMed]
  20. J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
    [CrossRef]
  21. B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
    [CrossRef]
  22. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
    [CrossRef]
  23. R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
    [CrossRef] [PubMed]
  24. L. Morales-Molina, R. A. Vicencio, “Trapping of discrete solitons by defects in nonlinear waveguide arrays,” Opt. Lett. 31, 966–968 (2006).
    [CrossRef]
  25. M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 33, 2332–2334 (2006).
    [CrossRef]
  26. M. I. Molina, Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
    [CrossRef] [PubMed]
  27. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
    [CrossRef]
  28. A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
    [CrossRef]
  29. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
  30. S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
    [CrossRef]
  31. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  32. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
    [CrossRef]
  33. S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
    [CrossRef]
  34. S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
    [CrossRef]
  35. A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
    [CrossRef]
  36. D. A. Zezyulin, V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
    [CrossRef]
  37. I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
    [CrossRef]
  38. K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
    [CrossRef]
  39. J. Pickton, H. Susanto, “Integrability of PT-symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
    [CrossRef]
  40. A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).
  41. T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
    [CrossRef]
  42. B. Maes, M. Soljačić, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006).
    [CrossRef]
  43. E. N. Bulgakov, A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010).
    [CrossRef]
  44. E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
    [CrossRef]
  45. M. I. Molina, G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993).
    [CrossRef]
  46. B. C. Gupta, K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
    [CrossRef]
  47. V. A. Brazhnyi, B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
    [CrossRef]
  48. Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
    [CrossRef]
  49. B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
    [CrossRef]
  50. M. Vakhitov, A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973).
    [CrossRef]
  51. L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
    [CrossRef]
  52. E. A. Kuznetsov, F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011).
    [CrossRef]
  53. M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
    [CrossRef]
  54. J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
    [CrossRef]
  55. H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
    [CrossRef]
  56. S. Longhi, “Invisibility in PT-symmetric complex crystals”, J. Phys. A: Math. Theor. 44, 485302 (2011).
    [CrossRef]
  57. Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
    [CrossRef]

2013 (6)

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
[CrossRef]

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

J. Pickton, H. Susanto, “Integrability of PT-symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
[CrossRef]

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[CrossRef]

2012 (6)

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[CrossRef]

2011 (6)

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

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

S. Longhi, “Invisibility in PT-symmetric complex crystals”, J. Phys. A: Math. Theor. 44, 485302 (2011).
[CrossRef]

V. A. Brazhnyi, B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[CrossRef]

E. A. Kuznetsov, F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011).
[CrossRef]

E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[CrossRef]

2010 (6)

E. N. Bulgakov, A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[CrossRef]

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

A. Szameit, S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010).
[CrossRef]

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

2009 (2)

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[CrossRef]

2008 (5)

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

M. I. Molina, Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
[CrossRef] [PubMed]

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

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

2006 (6)

2005 (1)

A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

2004 (3)

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

2003 (5)

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

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

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[CrossRef]

2002 (2)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
[CrossRef]

2001 (2)

B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

2000 (1)

M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
[CrossRef]

1999 (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

1998 (1)

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[CrossRef]

1997 (1)

B. C. Gupta, K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
[CrossRef]

1993 (1)

M. I. Molina, G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993).
[CrossRef]

1973 (1)

M. Vakhitov, A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973).
[CrossRef]

Abdullaev, F. K.

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

Achilleos, V.

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Alexander, T. J.

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Baets, R.

Barashenkov, I. V.

I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

Bender, C. M.

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

Bergé, L.

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[CrossRef]

Bersch, C.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

Bienstman, P.

Brazhnyi, V. A.

V. A. Brazhnyi, B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[CrossRef]

Bulgakov, E.

E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[CrossRef]

Bulgakov, E. N.

E. N. Bulgakov, A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010).
[CrossRef]

Burghoff, J.

Chen, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Chen, Z.

Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Chiofalo, M. L.

M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
[CrossRef]

Chow, K. W.

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Christodoulides, D. N.

Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[CrossRef]

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

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

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

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

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

Cohen, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

Denz, C.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Desyatnikov, A. S.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Dias, F.

E. A. Kuznetsov, F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011).
[CrossRef]

Ding, E.

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Dmitriev, S. V.

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Eisenberg, H. S.

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

El-Ganainy, R.

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

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

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

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

Ellis, F. M.

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Eugenieva, E. D.

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Flach, S.

I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Fleischer, J. W.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Frantzeskakis, D. J.

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

Freedman, B.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

Gaididei, Y. B.

P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
[CrossRef]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 33, 2332–2334 (2006).
[CrossRef]

Gorza, S.-P.

Gupta, B. C.

B. C. Gupta, K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
[CrossRef]

Haelterman, M.

Hudock, J.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

Hukriede,

Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[CrossRef]

Jackson, G. S.

I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
[CrossRef]

Joannopoulos, J. D.

Johansson, M.

M. Öster, M. Johansson, “Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2,” Phys. Rev. E 73, 066608 (2006).
[CrossRef]

Kaiser, F.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Kevrekidis, P. G.

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Kip, D.

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

Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[CrossRef]

Kivshar, Y. S.

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

M. I. Molina, Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
[CrossRef] [PubMed]

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 33, 2332–2334 (2006).
[CrossRef]

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

Kolokolov, A.

M. Vakhitov, A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973).
[CrossRef]

Konotop, V. V.

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

Kottos, T.

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Królikowski, W.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Kundu, K.

B. C. Gupta, K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov, F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Lai, S. K.

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006).
[CrossRef]

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

Li, K.

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

Li, Y.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Lifshitz, R.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

Lin, Z.

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Longhi, S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

S. Longhi, “Invisibility in PT-symmetric complex crystals”, J. Phys. A: Math. Theor. 44, 485302 (2011).
[CrossRef]

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[CrossRef]

Maes, B.

Makasyuk, I.

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Makris, K. G.

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

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, 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, Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Malomed, B. A.

T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

V. A. Brazhnyi, B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Mandelik, D.

Manela, O.

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

Martin, H.

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Mayteevarunyoo, T.

T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[CrossRef]

Mejía-Cortés, C.

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

Miri, M. A.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

Modotto, D.

Molina, M. I.

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

M. I. Molina, Y. S. Kivshar, “Nonlinear localized modes at phase-slip defects in waveguide arrays,” Opt. Lett. 33, 917–919 (2008).
[CrossRef] [PubMed]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 33, 2332–2334 (2006).
[CrossRef]

M. I. Molina, G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993).
[CrossRef]

Morales-Molina, L.

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Muga, J. G.

A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

Musslimani, Z. H.

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

Näger, J.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

Neshev, D.

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Neshev, D. N.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Nolte, S.

Öster, M.

M. Öster, M. Johansson, “Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2,” Phys. Rev. E 73, 066608 (2006).
[CrossRef]

Ostrovskaya, E. A.

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

Pang, W.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Pertsch, T.

Peschel, U.

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Pichugin, K. N.

E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[CrossRef]

Pickton, J.

J. Pickton, H. Susanto, “Integrability of PT-symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

Ramezani, H.

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

Regensburger, A.

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

Richter, T.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Rodrigues, A. S.

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

Roeksabutr, A.

T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[CrossRef]

Runde, D.

Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

Rüter, C. E.

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

Sadreev, A.

E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[CrossRef]

Sadreev, A. F.

E. N. Bulgakov, A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010).
[CrossRef]

Schindler, J.

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

Sears, S.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[CrossRef]

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

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

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

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Soljacic, M.

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

R. Morandotti, H. S. Eisenberg, D. Mandelik, Y. Silberberg, D. Modotto, M. Sorel, C. R. Stanley, J. S. Aitchison, “Interactions of discrete solitons with structural defects,” Opt. Lett. 28, 834–836 (2003).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

Stanley, C. R.

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

Succi, S.

M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
[CrossRef]

Suchkov, S. V.

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

Sukhorukov, A. A.

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

M. I. Molina, I. L. Garanovich, A. A. Sukhorukov, Y. S. Kivshar, “Discrete surface solitons in semi-infinite binary waveguide arrays,” Opt. Lett. 33, 2332–2334 (2006).
[CrossRef]

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

Sukhorukova, A. A.

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

Susanto, H.

J. Pickton, H. Susanto, “Integrability of PT-symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
[CrossRef]

Szameit, A.

Terhalle, B.

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Tosi, M. P.

M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
[CrossRef]

Tsironis, G. P.

M. I. Molina, G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993).
[CrossRef]

Tünnermann, A.

Vakhitov, M.

M. Vakhitov, A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973).
[CrossRef]

Vicencio, R.

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

Vicencio, R. A.

Yang, J.

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[CrossRef]

Yu, Z.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zezyulin, D. A.

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

Zhang, H.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Zhou, J.

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Europhys. Lett. (1)

S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003).
[CrossRef]

Int. J. Theor. Phys. (1)

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

J. Comput. Phys. (1)

J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009).
[CrossRef]

J. Phys. A (1)

A. Ruschhaupt, F. Delgado, J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
[CrossRef]

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

S. Longhi, “Invisibility in PT-symmetric complex crystals”, J. Phys. A: Math. Theor. 44, 485302 (2011).
[CrossRef]

J. Phys. B: At. Mol. Opt. Phys. (1)

A. Szameit, S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010).
[CrossRef]

J. Phys. D (1)

Hukriede, D. Runde, D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003).
[CrossRef]

Nat. Phys. (1)

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

Nature (4)

T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef]

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

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rep. (4)

I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008).
[CrossRef]

L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998).
[CrossRef]

E. A. Kuznetsov, F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011).
[CrossRef]

Phys. Rev. A (10)

Z. Lin, J. Schindler, F. M. Ellis, T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012).
[CrossRef]

H. Ramezani, T. Kottos, R. El-Ganainy, D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010).
[CrossRef]

V. A. Brazhnyi, B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011).
[CrossRef]

C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010).
[CrossRef]

Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009).
[CrossRef]

S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011).
[CrossRef]

I. V. Barashenkov, G. S. Jackson, S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013).
[CrossRef]

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, F. K. Abdullaev, “PT-symmetric coupler with χ(2) nonlinearity,” Phys. Rev. A 88, 053820 (2013).
[CrossRef]

J. Pickton, H. Susanto, “Integrability of PT-symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
[CrossRef]

Phys. Rev. B (4)

E. N. Bulgakov, A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010).
[CrossRef]

E. Bulgakov, A. Sadreev, K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011).
[CrossRef]

M. I. Molina, G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993).
[CrossRef]

B. C. Gupta, K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997).
[CrossRef]

Phys. Rev. E (8)

T. Mayteevarunyoo, B. A. Malomed, A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[CrossRef]

M. L. Chiofalo, S. Succi, M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012).
[CrossRef]

B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002).
[CrossRef]

P. G. Kevrekidis, B. A. Malomed, Z. Chen, D. J. Frantzeskakis, “Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 70, 056612 (2004).
[CrossRef]

M. Öster, M. Johansson, “Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2,” Phys. Rev. E 73, 066608 (2006).
[CrossRef]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Phys. Rev. Lett. (8)

D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001).
[CrossRef] [PubMed]

D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D. N. Christodoulides, “Observation of vortex-ring discrete solitons in 2D photonic lattices,” Phys. Rev. Lett. 92, 123904 (2004).
[CrossRef]

B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef] [PubMed]

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

A. Regensburger, M. A. Miri, C. Bersch, J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

Radiophys. Quantum Electron (1)

M. Vakhitov, A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973).
[CrossRef]

Rep. Prog. Phys. (1)

Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[CrossRef]

Rom. Rep. Phys. (1)

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, C. M. Bender, “PT-symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

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Figures (8)

Fig. 1
Fig. 1

The structure of the system. A pair of nonlinear cores, carrying mutually balanced gain and loss, are inserted into the host array of linear waveguides with coupling constant C0. The strength of the coupling between the inserted elements, Cd, is used as a basic control parameter.

Fig. 2
Fig. 2

(a) The blue (even) and red (odd) profiles display the real and imaginary parts of the stationary lattice field, Un [see Eq. (6)], of a stable fundamental PT soliton for (Cd, λ, κ) = (0.5, 0.5, 0.2), corresponding to ϕ+ and A + 2 in Eqs. (9) and (10). The total power of this soliton is P = 1.1583, see Eq. (15). (b,c) Eigenvalues of small perturbations around this soliton and direct simulations of the perturbed evolution prove that it is stable. (d) An unstable fundamental PT soliton solution for (Cd, λ, κ) = (0.8, 0.5, 0.2), with total power P = 1.6402. (e,f) Eigenvalues and direct simulations demonstrate that the latter soliton is unstable.

Fig. 3
Fig. 3

(a) A stable dipole-mode PT soliton for (Cd, λ, κ) = (0.5, 2.5, 0.3), corresponding to ϕ and A 2 in Eqs. (9) and (10). The total power of this soliton is P = 13.071, see Eq. (15). (b,c) Eigenvalues and direct simulations demonstrate that this soliton is stable. (d) An unstable dipole PT soliton, found for (Cd, λ, κ) = (0.5, 1.5, 0.3), with total power P = 5.5584. (e,f) Eigenvalue and direct simulations demonstrate instability of the latter soliton.

Fig. 4
Fig. 4

Plots of the total power, P+(λ) and P(λ) (black and dashed red curves, respectively), defined as per Eq. (15), for tightly [(κ, Cd) = (0.2, 0.5)] (a) and loosely [(κ, Cd) = (2, 0.4)] (b) knit lattices, which are identified pursuant to Eq. (13). Recall that P+(λ) and P(λ) pertain, respectively, to the fundamental and dipole-mode discrete solitons.

Fig. 5
Fig. 5

(a1)–(a4) The existence and stability diagram for the fundamental solitons in the plane of (λ, κ) for Cd/C0 = 0.85 (a1), Cd/C0 = 1 (a2), Cd/C0 = 2 (a3), and Cd/C0 = 4 (a4) [recall that λ determines the localization of the solitons as per Eqs. (6) and (7), κ is the gain/loss coefficient of the PT -symmetric defect, and Cd is the coefficient of the coupling between the two cores forming the defect]. The solitons do not exist in the white regions, viz., exactly as follows from the analytical solution, at κ > Cd (at the top of the panels) and at λ < λmin(κ) [see Eq. (14)], i.e., in the left bottom corners of panels (a3) and (a4). In the gray area, the solitons exist but do not satisfy the VK criterion [see Eq. (17)], i.e., they are definitely unstable. The computation of the stability eigenvalues, based on Eq. (5), demonstrates that the solitons are completely stable in the red area, but unstable against perturbation modes not covered by the VK criterion in the yellow area. The white, gray and red/yellow colors have the same meaning in existence and stability diagrams displayed in other figures below. The horizontal dotted lines in panel (a3) and (a4) designate the border between regions of the tightly and loosely knit linear lattices, defined as per Eq. (13), the lattices being “loose” [i.e., with λmin > 0, see Eq. (14)] beneath these lines. In panels (a1) and (a2), which correspond to C0Cd, the linear lattices may only be tightly knit ones. (b1)-(b4) The stability and instability areas from (a1)–(a4) for the fundamental solitons, replotted in the plane of (P, κ) at same values of Cd.(c1)–(c4) The same stability diagrams as in (b1)–(b4), but produced by direct simulations. In the white area, the imaginary-time simulations do not converge to stationary solitons.

Fig. 6
Fig. 6

(a) Areas of the stability (red) and instability (yellow) for the fundamental discrete solitons in the (P, Cd/C0) plane at a fixed value of the gain-loss coefficient, κ = 0.01. (b) Regions of the existence of stable symmetric (red) and asymmetric (yellow) fundamental discrete solitons in the same system with κ = 0 (no gain and loss).

Fig. 7
Fig. 7

Stability regions for the dipole-mode solitons with Cd = 0.85 (a), Cd = 1 (b), Cd = 2 (c) and Cd = 4 (d) in the (P, κ) plane. The meaning of the colors is the same as in other figures: red for stable, and yellow for unstable.

Fig. 8
Fig. 8

Numerically obtained solutions of the scattering problem are shown in the plane of the gain-loss coefficient, κ, and wavenumber of the incident wave, q. Panels (a,b,c,d) show the reflection coefficient, |R|/I2, for I = 0.1 (a), 0.5 (b), 1 (c) and 10 (d). (e,f,g,h): The transmission coefficient, |T|/I2, for I = 0.1 (e), 0.5 (f), 1 (g) and 10 (h). (i,j,k,l): The sum of reflection and transmission coefficients, (|R|2 + |T|2)/I2, for I = 0.1 (i), 0.5 (j), 1 (k) and 10 (l). In these figures, we fix Cd = C0 = 0.5.

Equations (21)

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i d u n d z = ( C n , n 1 u n 1 + C n + 1 , n u n + 1 ) γ n | u n | 2 u n + i κ n u n .
C n , n 1 = { C d , n = N / 2 C 0 , n N / 2 , γ n = { γ , n = N / 2 , N / 2 + 1 0 , n N / 2 , N / 2 + 1 , κ n = { κ , n = N / 2 κ n = N / 2 + 1 0 , n N / 2 , N / 2 + 1 ,
P = n = 1 N | u n | 2 .
u n ( z ) = U n e i K z ,
u n = e i K z ( U n + w n e i G z + v n * e i G * z ) ,
( C + K + 2 γ n | U n | 2 + i κ n γ n U n 2 γ n U n * 2 C K 2 γ n | U n | 2 + i κ n ) ( w v ) = G ( w v ) .
U n = { A e i ϕ / 2 exp [ λ ( N / 2 n ) ] , at n N / 2 A e i ϕ / 2 exp [ λ ( n 1 N / 2 ) ] , at n N / 2 + 1 ,
K = 2 C 0 cosh ( λ )
K = C 0 e λ + C d e + i ϕ + A 2 i κ , K = C 0 e λ + C d e i ϕ + A 2 + i κ ,
ϕ + = arcsin ( κ / C d ) , ϕ = π arcsin ( κ / C d ) ,
A ± 2 = C 0 e λ C d 2 κ 2 ,
A 2 ( λ = 0 ) = C 0 + C d 2 κ 2 .
A + 2 ( λ = 0 ) = C 0 C d 2 κ 2 ,
C 0 > C 0 ( 0 ) C d 2 κ 2
λ λ min = ( 1 / 2 ) ln ( C d 2 κ 2 ) ln C 0 .
P ± = 2 C 0 e λ C d 2 κ 2 1 e 2 λ .
C 0 ( e λ 0 ) 3 3 C 0 e λ 0 ± 2 C d 2 κ 2 = 0 .
Δ VK C 0 ( e 3 λ 3 e λ ) ± 2 C d 2 κ 2 > 0 ,
U n = { I e i q ( n N / 2 ) + Re i q ( n N / 2 ) , n N / 2 T e i q [ n ( N / 2 + 1 ) ] , n N / 2 + 1 .
K = 2 cos ( 2 q ) ,
2 C 0 cos ( 2 q ) ( I + R ) = C 0 ( e i q I + e i q R ) + C d T | T + R | 2 ( T + R ) i κ ( I + R ) 2 C 0 cos ( 2 q ) T = C 0 T e i q + C d ( I + R ) + | T | 2 T + i κ T

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