Abstract

We address the properties of arc-shaped solitons supported by defocusing nonlinearity on a partially-parity-time symmetric ring, including the existence and stability. Four types of arc-shaped solitons are found. The existence region of arc-shaped solitons with two or more bright spots is the same, while it is slightly smaller in value than that of fundamental solitons. Also, the existence domains of arc-shaped solitons shrink with the increase of the strength of the gain and loss term. At moderate gain and loss levels, stable arc-shaped solitons are usually localized in the middle of their existence domain. The characteristics of unstable arc-shaped solitons are considered to be related to the power-flow of the solitons, because the sidelobes of solitons extend to multiple Gaussian waveguides at both ends of their existence, and then not all the power-flows in each Gaussian waveguide flow from the gain to the loss region. Otherwise, robust nonlinear arc-shaped states with four different bright spots can be excited by Gaussian beams. This work offers us new insight and understanding of optical solitons on a partially-parity-time symmetric ring.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2019 (1)

2018 (1)

2017 (1)

2016 (4)

C. Huang and L. Dong, “Stable vortex solitons in a ring-shaped partially-𝒫𝒯-symmetric potential,” Opt. Lett. 41, 5194–5197 (2016).
[Crossref] [PubMed]

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

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

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

2015 (1)

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological states in partially-𝒫𝒯-symmetric azimuthal potentials,” Phys. Rev. Lett. 115, 193902 (2015).
[Crossref]

2014 (2)

2013 (2)

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

C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a 𝒫𝒯 symmetry,” Opt. Express 21, 3917–3925 (2013).
[Crossref] [PubMed]

2012 (3)

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

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

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

2011 (2)

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

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 (1)

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

2009 (2)

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

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

2008 (2)

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

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. 33, 2949–2951 (2008).
[Crossref] [PubMed]

2007 (2)

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

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

1999 (1)

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

1998 (1)

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

Abdullaev, F. K.

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

Aimez, V.

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

Bartal, G.

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

Bender, C. M.

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

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

Boettcher, S.

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

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

Chen, X.

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

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

Christodoulides, D.

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

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

Christodoulides, D. N.

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

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

Dong, L.

Duchesne, D.

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

El-Ganainy, R.

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

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

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

Fishman, S.

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

Ge, L.

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

Guo, A.

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

He, Y.

X. Zhu and Y. He, “Vector solitons in nonparity-time-symmetric complex potentials,” Opt. Express 26, 26511–26519 (2018).
[Crossref] [PubMed]

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

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

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

Huang, C.

Kartashov, Y. V.

Kip, D.

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

Kivshar, Y. S.

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]

Konotop, V. V.

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

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological states in partially-𝒫𝒯-symmetric azimuthal potentials,” Phys. Rev. Lett. 115, 193902 (2015).
[Crossref]

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

Li, C.

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

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

Longhi, S.

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

Makris, K.

Makris, K. G.

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

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

Malomed, B. A.

Meisinger, P. N.

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

Mihalache, D.

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

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

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

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).
[Crossref]

Morandotti, R.

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

Musslimani, Z. H.

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

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

Nixon, S.

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

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

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

Rüter, C. E.

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

Salamo, G.

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

Schwartz, T.

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

Segev, M.

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

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

Siviloglou, G.

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

Torner, L.

Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological states in partially-𝒫𝒯-symmetric azimuthal potentials,” Phys. Rev. Lett. 115, 193902 (2015).
[Crossref]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. 33, 2949–2951 (2008).
[Crossref] [PubMed]

Volatier-Ravat, M.

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

Vysloukh, V. A.

Yang, J.

J. Yang, “Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions,” Opt. Lett. 42, 4067–4070 (2017).
[Crossref] [PubMed]

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

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

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

J. Yang, “Partially 𝒫𝒯 symmetric optical potentials with all-real spectra and soliton families in multidimensions,” Opt. Lett. 39, 1133–1136 (2014).
[Crossref] [PubMed]

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[Crossref]

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[Crossref]

Ye, F.

Zezyulin, D. A.

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

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

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in 𝒫𝒯-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012).
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Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Phys. Lett. A (1)

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

Phys. Rev. A (6)

S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016).
[Crossref]

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

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

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

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

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Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological states in partially-𝒫𝒯-symmetric azimuthal potentials,” Phys. Rev. Lett. 115, 193902 (2015).
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Rev. Mod. Phys. (1)

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

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1 Profiles of modulus (a), real (b) and imaginary parts (c) of a ring-shaped gain-loss potential. pr = 16, pi = 7.466, ϱ = 0.1 and x, y ∈ [−9, +9] in all panels.
Fig. 2
Fig. 2 Field moduli of arc-shaped solitons with 1 ∼ 4 spots are plotted in (a–d). pr = 16, pi = 7.466, ϱ = 0.1, β = 4.0 in all panels.
Fig. 3
Fig. 3 The real parts (a1–a4), imaginary parts (b1–b4), and absolute value of the imaginary parts (c1–c4) of the arc-shaped solitons with β = 4.0. The modulus (a5) of a arc-shaped soliton at a small propagation constant β = 1.1 and the absolute value of the linear mode (b5) are depicted. pr = 16, ϱ = 0.1 in all panels.
Fig. 4
Fig. 4 Dependencies of power of arc-shaped solitons with 1 ∼ 4 bright spots on β for the relative magnitude of gain-loss term ϱ = 0.1 (a) and 0.3 (b). The black and red segments represent stable and unstable regions, respectively. In the green segments, the corresponding solitons radiate some energy and convert to another robust solutions. Existence domains of the fundamental arc-shaped solitons (c) in the (ϱ, β) plane. (d) Effective width of arc-shaped solitons versus β for ϱ = 0.1. Power versus propagation distance z for the fundamental arc-shaped solitons at β = 4.8 (e) and β = 5.0 (f), for the even bound states at β = 2.5 (g) and β = 5.2 (h). ϱ = 0.3 in (e–h), pr = 16 in all panels.
Fig. 5
Fig. 5 Transverse power-flow vector S⃗ of arc-shaped solitons with 1 ∼ 4 bright spots. The arrow indicates the direction of S⃗. β = 4.0, pr = 16, ϱ = 0.3 in all panels.
Fig. 6
Fig. 6 Stable (a–d) and unstable (e–h) propagation dynamics of arc-shaped solitons with 1 ∼ 4 bright spots. β = 4.8 (a), 5.2 (b), 3.5 (c), 4.7 (d), 5.0 (e), 2.7 (f), 2.5 (g), and 0.1 (h). ϱ = 0.3 in panels (a), (b), (e), (f) and (g). ϱ = 0.1 in panels (c), (d), and (h). pr = 16 in all panels.
Fig. 7
Fig. 7 The dependencies of power U on the propagation distance z are plotted in (a–d), and four different types of nonlinear arc-shaped states at z = 5000 are illustrated in (A–D). (xk ; yk)=(0; −5.4) in (a), (xk ; yk)=(−5.318, −5.318; −0.9377, 0.9377) in (b), (xk ; yk)=(−1.847, 0, 1.847; −5.074, −5.4, −5.074) in (c), (xk ; yk)=(−4.677, −5.318, −5.318, −4.677; −2.7, −0.9377, 0.9377, 2.7) in (d), A = 2 in all panels.

Equations (4)

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i Ψ z = 1 2 ( 2 Ψ x 2 + 2 Ψ y 2 ) V Ψ + | Ψ | 2 Ψ ,
V re = k = 1 N exp [ ( x x k ) 2 / d 2 ( y y k ) 2 / d 2 ] ,
V im = k = 1 N exp [ ( x x k ) 2 / d 2 ( y y k ) 2 / d 2 ] × [ y cos ( ϕ k ) x sin ( ϕ k ) ] ,
1 2 ( 2 ϕ x 2 + 2 ϕ y 2 ) + V ϕ β ϕ | ϕ | 2 ϕ = 0 .

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