Abstract

We introduce a 2D network built of 𝒫𝒯 -symmetric dimers with on-site cubic nonlinearity, the gain and loss elements of the dimers being linked by parallel square-shaped lattices. The system may be realized as a set of 𝒫𝒯 -symmetric dual-core waveguides embedded into a photonic crystal. The system supports 𝒫𝒯 -symmetric and antisymmetric fundamental solitons (FSs) and on-site-centered solitary vortices (OnVs). Stability of these discrete solitons is the central topic of the consideration. Their stability regions in the underlying parameter space are identified through the computation of stability eigenvalues, and verified by direct simulations. Symmetric FSs represent the system’s ground state, being stable at lowest values of the power, while anti-symmetric FSs and OnVs are stable at higher powers. Symmetric OnVs, which are also stable at lower powers, are remarkably robust modes: on the contrary to other 𝒫𝒯 -symmetric states, unstable OnVs do not blow up, but spontaneously rebuild themselves into stable FSs.

© 2014 Optical Society of America

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2014 (8)

B. A. Malomed, “Spatial solitons supported by localized gain [Invited],” J. Opt. Soc. Am. B 31, 2460–2475 (2014).
[Crossref]

J. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 30, 5547–5550 (2014).
[Crossref]

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

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

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

C. H. Liang, D. D. Scott, and Y. N. Joglekar, “𝒫𝒯 restoration via increased loss and gain in the 𝒫𝒯 -symmetric Aubry-André model,” Phys. Rev. A 89, 030102(R) (2014).
[Crossref]

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete Solitons in Waveguide Arrays with Long-Range Linearly Coupled Effect,” J. Phys. Soc. Jpn. 83, 034404 (2014).
[Crossref]

2013 (14)

D. Leykam, V. V Konotop, and A. S. Desyatnikov, “Discrete vortex solitons and parity time symmetry,” Opt. Lett. 38, 371 (2013).
[Crossref] [PubMed]

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

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, “Nonlinear stationary states in 𝒫𝒯 -symmetric lattices,” SIAM J. Appl. Dyn. Syst. 12, 1210–1236 (2013).
[Crossref]

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

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

K. Li, D. A. Zezyulin, V. V. Konotop, and P. G. Kevrekidis, “Parity-time-symmetric optical coupler with birefringent arms,” Phys. Rev. A 87, 033812 (2013).
[Crossref]

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

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

J. Pickton and 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, and C. M. Bender, “𝒫𝒯 -symmetric double-well potentials revisited: bifurcations, stability and dynamics,” Rom. Rep. Phys. 65, 5–26 (2013).

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

Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
[Crossref]

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

2012 (10)

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[Crossref]

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

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, “Beam propagation in gain-loss balanced waveguides,” Opt. Commun. 285, 3441–3444 (2012).
[Crossref]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85063837 (2012).
[Crossref]

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear 𝒫𝒯 -symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

F. C. Moreira, F. Kh. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with 𝒫𝒯 -symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
[Crossref]

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

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of 𝒫𝒯 -symmetric couplers,” Phys. Rev. A 85, 033835 (2012).
[Crossref]

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

V. V. Konotop, D. E. Pelinovsky, and D. A. Zezyulin, “Discrete solitons in 𝒫𝒯 -symmetric lattices,” EPL 100, 56006 (2012).
[Crossref]

2011 (10)

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E, 64, 026601 (2011).
[Crossref]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and 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. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
[Crossref]

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

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

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

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

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

K. Li and P. G. Kevrekidis, “𝒫𝒯 -symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly 𝒫𝒯 -symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[Crossref]

2010 (2)

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, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Physics 6, 192 (2010).
[Crossref]

2009 (3)

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

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

M. Öster and M. Johansson, “Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling,” Physica D 238, 88–99 (2009).
[Crossref]

2008 (3)

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

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

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

2007 (2)

H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[Crossref]

G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
[Crossref]

2006 (1)

R. A. Vicencio and M. Johansson, “Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E 73, 046602 (2006).
[Crossref]

2005 (1)

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

2004 (1)

W. Z. Bao and Q. Du, “Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow,” SIAM J. Sci. Comp. 25, 1674–1697 (2004).
[Crossref]

2002 (1)

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

2000 (1)

M. L. Chiofalo, S. Succi, and 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]

1998 (2)

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

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

1997 (1)

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341 (1997).
[Crossref]

1973 (1)

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

Abdullaev, F. Kh.

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

F. C. Moreira, F. Kh. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with 𝒫𝒯 -symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
[Crossref]

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, “Beam propagation in gain-loss balanced waveguides,” Opt. Commun. 285, 3441–3444 (2012).
[Crossref]

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

Achilleos, V.

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

Aimez, V.

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

Aitchison, J. S.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341 (1997).
[Crossref]

Alexeeva, N. V.

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85063837 (2012).
[Crossref]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[Crossref]

Bao, W. Z.

W. Z. Bao and Q. Du, “Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow,” SIAM J. Sci. Comp. 25, 1674–1697 (2004).
[Crossref]

Barashenkov, I. V.

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

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[Crossref]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85063837 (2012).
[Crossref]

Bender, C. M.

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

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

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

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, and J. Näger, “Observation of defect states in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (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]

Brody, D. C.

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

G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
[Crossref]

H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[Crossref]

Chai, J.

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Chen, L.

Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
[Crossref]

Chen, Z.

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

Chiofalo, M. L.

M. L. Chiofalo, S. Succi, and 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]

Christodoulides, D. N.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[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,” Nature Physics 6, 192 (2010).
[Crossref]

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

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

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

Delgado, F.

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

Desyatnikov, A. S.

Dmitriev, S. V.

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

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of 𝒫𝒯 -symmetric couplers,” Phys. Rev. A 85, 033835 (2012).
[Crossref]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and 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. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
[Crossref]

Driben, R.

Du, Q.

W. Z. Bao and Q. Du, “Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow,” SIAM J. Sci. Comp. 25, 1674–1697 (2004).
[Crossref]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[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,” Nature Physics 6, 192 (2010).
[Crossref]

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

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

Flach, S.

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

Frantzeskakis, D. J.

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

G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
[Crossref]

H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[Crossref]

Fu, S.

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete Solitons in Waveguide Arrays with Long-Range Linearly Coupled Effect,” J. Phys. Soc. Jpn. 83, 034404 (2014).
[Crossref]

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]

Günther, U.

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear 𝒫𝒯 -symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

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

Guo, A.

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

Guo, X.

Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
[Crossref]

He, Y.

Herring, G.

G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
[Crossref]

Hua, S.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Huang, J.

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

Hutchings, D. C.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341 (1997).
[Crossref]

Jackson, G. S.

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

Jiang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Jiang, X.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
[Crossref]

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

Joglekar, Y. N.

C. H. Liang, D. D. Scott, and Y. N. Joglekar, “𝒫𝒯 restoration via increased loss and gain in the 𝒫𝒯 -symmetric Aubry-André model,” Phys. Rev. A 89, 030102(R) (2014).
[Crossref]

Johansson, M.

M. Öster and M. Johansson, “Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling,” Physica D 238, 88–99 (2009).
[Crossref]

R. A. Vicencio and M. Johansson, “Discrete soliton mobility in two-dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E 73, 046602 (2006).
[Crossref]

Jones, H. F.

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

Kang, J. U.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341 (1997).
[Crossref]

Kartashov, Y. V.

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

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

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, “Guided modes and symmetry breaking supported by localized gain,” in: Spontaneous Symmetry Breaking, Self-Trapping, and Josephson OscillationsB. A. Malomed, ed. (Springer2013).

Kevrekidis, P. G.

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

P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, “Nonlinear stationary states in 𝒫𝒯 -symmetric lattices,” SIAM J. Appl. Dyn. Syst. 12, 1210–1236 (2013).
[Crossref]

A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and 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, V. V. Konotop, and P. G. Kevrekidis, “Parity-time-symmetric optical coupler with birefringent arms,” Phys. Rev. A 87, 033812 (2013).
[Crossref]

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear 𝒫𝒯 -symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E, 64, 026601 (2011).
[Crossref]

K. Li and P. G. Kevrekidis, “𝒫𝒯 -symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
[Crossref]

H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
[Crossref]

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations, and Physical Perspectives (Springer2009).
[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,” Nature Physics 6, 192 (2010).
[Crossref]

Kivshar, Y. S.

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of 𝒫𝒯 -symmetric couplers,” Phys. Rev. A 85, 033835 (2012).
[Crossref]

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

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85063837 (2012).
[Crossref]

N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, “Optical solitons in 𝒫𝒯 -symmetric nonlinear couplers with gain and loss,” Phys. Rev. A 85, 063837 (2012).
[Crossref]

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly 𝒫𝒯 -symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
[Crossref]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
[Crossref]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and 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]

Klaiman, S.

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

Kolokolov, A. A.

N. G. Vakhitov and A. 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

Konotop, V. V.

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

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

K. Li, D. A. Zezyulin, V. V. Konotop, and P. G. Kevrekidis, “Parity-time-symmetric optical coupler with birefringent arms,” Phys. Rev. A 87, 033812 (2013).
[Crossref]

F. C. Moreira, F. Kh. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with 𝒫𝒯 -symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
[Crossref]

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

V. V. Konotop, D. E. Pelinovsky, and D. A. Zezyulin, “Discrete solitons in 𝒫𝒯 -symmetric lattices,” EPL 100, 56006 (2012).
[Crossref]

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

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and D. A. Zezyulin, “Guided modes and symmetry breaking supported by localized gain,” in: Spontaneous Symmetry Breaking, Self-Trapping, and Josephson OscillationsB. A. Malomed, ed. (Springer2013).

Lai, T.

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

Lazarides, N.

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

Lee, C.

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

Leykam, D.

Li, G.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
[Crossref]

Li, H.

Li, K.

K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, and F. Kh. 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, and 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, V. V. Konotop, and P. G. Kevrekidis, “Parity-time-symmetric optical coupler with birefringent arms,” Phys. Rev. A 87, 033812 (2013).
[Crossref]

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear 𝒫𝒯 -symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

K. Li and P. G. Kevrekidis, “𝒫𝒯 -symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics,” Phys. Rev. E 83, 066608 (2011).
[Crossref]

Li, Y.

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete Solitons in Waveguide Arrays with Long-Range Linearly Coupled Effect,” J. Phys. Soc. Jpn. 83, 034404 (2014).
[Crossref]

Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
[Crossref]

Liang, C. H.

C. H. Liang, D. D. Scott, and Y. N. Joglekar, “𝒫𝒯 restoration via increased loss and gain in the 𝒫𝒯 -symmetric Aubry-André model,” Phys. Rev. A 89, 030102(R) (2014).
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S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010).
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S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
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Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced 𝒫𝒯 transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
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Mai, Z.

Z. Mai, S. Fu, J. Wu, and Y. Li, “Discrete Solitons in Waveguide Arrays with Long-Range Linearly Coupled Effect,” J. Phys. Soc. Jpn. 83, 034404 (2014).
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Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Physics 6, 192 (2010).
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K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Beam dynamics in 𝒫𝒯 symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
[Crossref]

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

Malomed, B. A.

X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, “Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear 𝒫𝒯 -symmetric defect,” Opt. Exp. 22, 6055 (2014).
[Crossref]

H. Li, X. Zhu, Z. Shi, B. A. Malomed, T. Lai, and C. Lee, “Bulk vortex and horseshoe surface modes in parity-time-symmetric media,” Phys. Rev. A 89, 053811 (2014).
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B. A. Malomed, “Spatial solitons supported by localized gain [Invited],” J. Opt. Soc. Am. B 31, 2460–2475 (2014).
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Y. V. Kartashov, B. A. Malomed, and L. Torner, “Unbreakable 𝒫𝒯 symmetry of solitons supported by inhomogeneous defocusing nonlinearity,” Opt. Lett. 39, 5641–5644 (2014).
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F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in 𝒫𝒯 -symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).
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T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
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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]

K. Li, P. G. Kevrekidis, B. A. Malomed, and U. Günther, “Nonlinear 𝒫𝒯 -symmetric plaquettes,” J. Phys. A 45, 444021 (2012).
[Crossref]

S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of 𝒫𝒯 -symmetric couplers,” Phys. Rev. A 85, 033835 (2012).
[Crossref]

S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
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R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
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B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E, 64, 026601 (2011).
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A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly 𝒫𝒯 -symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
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G. Herring, P. G. Kevrekidis, B. A. Malomed, R. Carretero-González, and D. J. Frantzeskakis, “Symmetry breaking in linearly coupled dynamical lattices,” Phys. Rev. E 76, 066606 (2007).
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H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
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Mayteevarunyoo, T.

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

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

A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly 𝒫𝒯 -symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011).
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S. Klaiman, U. Günther, and N. Moiseyev, “Visualization of branch points in 𝒫𝒯 -symmetric waveguides,” Phys. Rev. Lett. 101, 080402 (2008).
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F. C. Moreira, V. V. Konotop, and B. A. Malomed, “Solitons in 𝒫𝒯 -symmetric periodic systems with the quadratic nonlinearity,” Phys. Rev. A 87, 013832 (2013).
[Crossref]

F. C. Moreira, F. Kh. Abdullaev, V. V. Konotop, and A. V. Yulin, “Localized modes in χ(2) media with 𝒫𝒯 -symmetric localized potential,” Phys. Rev. A 86, 053815 (2012).
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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯 -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
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K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “𝒫𝒯 symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
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A. Regensburger, M. A. Miri, C. Bersch, and J. Näger, “Observation of defect states in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
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V. V. Konotop, D. E. Pelinovsky, and D. A. Zezyulin, “Discrete solitons in 𝒫𝒯 -symmetric lattices,” EPL 100, 56006 (2012).
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Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced 𝒫𝒯 transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
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A. Regensburger, M. A. Miri, C. Bersch, and J. Näger, “Observation of defect states in 𝒫𝒯 -symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013).
[Crossref]

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A. S. Rodrigues, K. Li, V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis, and 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, and A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013).
[Crossref]

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A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of 𝒫𝒯 -symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005).
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C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Physics 6, 192 (2010).
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C. H. Liang, D. D. Scott, and Y. N. Joglekar, “𝒫𝒯 restoration via increased loss and gain in the 𝒫𝒯 -symmetric Aubry-André model,” Phys. Rev. A 89, 030102(R) (2014).
[Crossref]

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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,” Nature Physics 6, 192 (2010).
[Crossref]

Shi, Z.

H. Li, X. Zhu, Z. Shi, B. A. Malomed, T. Lai, and C. Lee, “Bulk vortex and horseshoe surface modes in parity-time-symmetric media,” Phys. Rev. A 89, 053811 (2014).
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H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36, 3290–3292 (2011).
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S. V. Suchkov, S. V. Dmitriev, B. A. Malomed, and Y. S. Kivshar, “Wave scattering on a domain wall in a chain of 𝒫𝒯 -symmetric couplers,” Phys. Rev. A 85, 033835 (2012).
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S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Scattering of the discrete solitons on the 𝒫𝒯 -symmetric defects,” Europhys. Lett. 100, 54003 (2012).
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S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, “Solitons in a chain of parity-time-invariant dimers,” Phys. Rev. E 84, 046609 (2011).
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S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a 𝒫𝒯 -symmetric defect,” Phys. Rev. A 84, 013833 (2011).
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S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Scattering of the discrete solitons on the 𝒫𝒯 -symmetric defects,” Europhys. Lett. 100, 54003 (2012).
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J. Pickton and H. Susanto, “Integrability of 𝒫𝒯 -symmetric dimers,” Phys. Rev. A 88, 063840 (2013).
[Crossref]

H. Susanto, P. G. Kevrekidis, R. Carretero-González, B. A. Malomed, and D. J. Frantzeskakis, “Mobility of discrete solitons in quadratically nonlinear media,” Phys. Rev. Lett. 99, 214103 (2007).
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N. Lazarides and G. P. Tsironis, “Gain-driven discrete breathers in𝒫𝒯 -symmetric nonlinear metamaterials,” Phys. Rev. Lett. 110, 053901 (2013).
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P. G. Kevrekidis, D. E. Pelinovsky, and D. Y. Tyugin, “Nonlinear stationary states in 𝒫𝒯 -symmetric lattices,” SIAM J. Appl. Dyn. Syst. 12, 1210–1236 (2013).
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L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nature Phot. 8, 524–529 (2014).
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Y. Li, X. Guo, L. Chen, C. Xu, J. Yang, X. Jiang, and M. Wang, “Coupled mode theory under the parity-time symmetry frame,” J. Lightwave Techn. 31, 2477–2481 (2013).
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K. Li, D. A. Zezyulin, V. V. Konotop, and P. G. Kevrekidis, “Parity-time-symmetric optical coupler with birefringent arms,” Phys. Rev. A 87, 033812 (2013).
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Phys. Rev. E (7)

G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in 𝒫𝒯 -symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013).
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Figures (9)

Fig. 1
Fig. 1

The transverse cross-section of the waveguiding system built as a juxtaposition of networks of active (red) and passive (blue) elements, which are coupled into ���� -symmetric dimers (the couplings are designated by short diagonal segments). The elements are connected into the networks by thin plates forming the double photonic crystal. The propagation axis (z) is directed perpendicular to the plane of the drawing.

Fig. 2
Fig. 2

(a) The cross-section, along m = 0, of a typical ���� -symmetric fundamental soliton produced by Eq. (1), with (C, γ) = (0.3, 0.4), and total power P = 2. Its real and imaginary parts precisely obey Eq. (7) with χ = χ+. Both the computation of stability eigenvalues and direct simulations demonstrate that this soliton is stable. (b) The cross-section (along m = 0) of a typical stable ���� -antisymmetric fundamental soliton, which is also stable, produced by stationary equations (1) for (C, γ) = (3, 0.5), and k = 10. The total power of this soliton is P = 33.8. (c) P(k) dependences for the symmetric (“SY”) and antisymmetric (“Anti-SY”) fundamental solitons, with C = 0.2 and γ = 3 / 2. The fitting result for them is P(k) = 4C + 2k ∓ 1, which is consistent with Eq. (11).

Fig. 3
Fig. 3

Here and in similar figures below, red and gray colors represent, respectively, stability and instability areas – here, for ���� -symmetric fundamental solitons – as found from the computation of the stability eigenvalues [see Eq. (5)], and verified by direct simulations. In white areas, soliton solutions cannot be found. The red-white and red-gray boundaries correspond, severally, to threshold values P th ( 1 ) and P th ( 2 ) (see the text).

Fig. 4
Fig. 4

(a,b) The stability area (red) of the antisymmetric fundamental solitons for (a) γ = 0.5 and (b) γ = 0.999. (c) The stability area in the (γ, C) plane for k = 10. (d) The stability diagram for γ = 0.5 in the (C, P) plane.

Fig. 5
Fig. 5

Real (a) and imaginary (b) parts, and the phase structure (c), of field um,n of a typical stable ���� -symmetric vortex soliton, for (C, γ) = (0.06, 0.4) and k = 1 (P = 1.65).

Fig. 6
Fig. 6

The same as in Fig. 5, but for a stable ���� -antisymmetric vortex, for (C, γ) = (2, 0.85) and k = 8 (P = 100.3). In this figure, both components are displayed, to stress that relation (7) for the antisymmetric vortex, with χ = χ, gives rise to rotation of the phase pattern by angle π – arcsin(0.85) ≈ 0.7π.

Fig. 7
Fig. 7

(a) The instability evolution of a symmetric vortex, with C = 0.06, γ = 0.4, and k = 0.96 (P = 1.37). This unstable vortex does not blow up, but spontaneously transforms into a stable ���� -symmetric fundamental soliton. In these panels, the two components of the vortex are juxtaposed, to display the evolution in cross section m = 0. (b) The instability development in an antisymmetric vortex, for C = 2, γ = 0.95, and k = 4 (P = 66.83). This unstable mode blows up in its active component, as is common for ���� -symmetric systems.

Fig. 8
Fig. 8

Stability and existence regions for ���� -symmetric vortex solitons: (a1) in the (k, C) plane, and (a2) in the (P, C) plane, for a fixed gain-loss coefficient, γ = 0.4. The variation of γ produces small changes in these diagrams (not shown here in detail). (b1,b2,c1) Stability regions of ���� -antisymmetric vortex solitons in the (k, C) plane: (a) for γ = 0.01, (b) for γ = 0.95, and (c1) for γ = 0.999. (c2) The stability region in the (P, C) plane for γ = 0.95. In this case, unlike what happens with symmetric vortices, the stability region strongly changes with the variation of γ.

Fig. 9
Fig. 9

The dependence of the total area of the stability regions in the (k, C) parameter plane on the gain-loss coefficient, γ, for the symmetric (a) and antisymmetric (b) vortex solitons. The continuous lines are only guides to the eye.

Tables (1)

Tables Icon

Table 1 Chemical potentials of the four coexisting soliton species at (C, γ) = (0.05, 0.1) and P = 1.9

Equations (14)

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i d d z ψ m , n = C 2 ( ψ m , n + 1 + ψ m , n 1 + ψ m 1 , n + ψ m + 1 , n 4 ψ m , n ) | ψ m , n | 2 ψ m , n φ m , n + i γ ψ m , n , i d d z φ m , n = C 2 ( φ m , n + 1 + φ m , n 1 + φ m 1 , n + φ m + 1 , n 4 φ m , n ) | φ m , n | 2 φ m , n ψ m , n i γ φ m , n .
P = m , n ( | ψ m , n | 2 + | φ m , n | 2 ) .
( k + i γ ) u m , n = C 2 ( u m , n + 1 + u m , n 1 + u m 1 , n + u m + 1 , n 4 u m , n ) + | u m , n | 2 u m , n + v m , n , ( k i γ ) v m , n = C 2 ( v m , n + 1 + v m , n 1 + v m 1 , n + v m + 1 , n 4 v m , n ) + | v m , n | 2 v m , n + u m , n .
L ^ U diag ( | U | 2 ) U = ( k + i Γ ^ ) U ,
( L ^ + k 2 diag ( | U | 2 ) + i Γ ^ diag ( U 2 ) diag ( U * 2 ) L ^ k + 2 diag ( | U | 2 ) + i Γ ^ ) ( α β ) = λ ( α β ) .
( k ± 1 γ 2 ) u m , n + C 2 ( u m , n + 1 + u m , n 1 + u m 1 , n + u m + 1 , n 4 u m , n ) + | u m , n | 2 u m , n = 0 ,
v m , n = e i χ ± u m , n , χ + = arcsin γ , χ = π arcsin γ ,
k eff = k 1 γ 2 ,
P > P th ( 1 ) = const C , const 5.7 .
P = 2 P 1 ( k 1 γ 2 ) .
P FS ( k , γ ) 2 ( 2 C + k 1 γ 2 ) .
{ u m , n , v m , n } on = { { U , V } at ( m , n ) = ( 0 , 1 ) , i { U , V } at ( m , n ) = ( 1 , 0 ) , { U , V } at ( m , n ) = ( 1 , 0 ) , i { U , V } at ( m , n ) = ( 0 , 1 ) , 0 , at all others ( m , n ) ,
{ u m , n , v m , n } off = { { U , V } at ( m , n ) = ( 0 , 0 ) , i { U , V } at ( m , n ) = ( 1 , 0 ) , { U , V } at ( m , n ) = ( 1 , 1 ) , i { U , V } at ( m , n ) = ( 0 , 1 ) , 0 , at all others ( m , n ) ,
P OnV = 4 P FS ( k ) ,

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