Abstract

We investigate the properties of spatial solitons in the fractional Schrödinger equation (FSE) with parity-time (PT)-symmetric lattice potential supported by the focusing of Kerr nonlinearity. Both one- and two-dimensional solitons can stably propagate in PT-symmetric lattices under noise perturbations. The domains of stability for both one- and two-dimensional solitons strongly depend on the gain/loss strength of the lattice. In the spatial domain, the solitons are rigidly modulated by the lattice potential for the weak diffraction in FSE systems. In the inverse space, due to the periodicity of lattices, the spectra of solitons experience sharp peaks when the values of wavenumbers are even. The transverse power flows induced by the imaginary part of the lattice are also investigated, which can preserve the internal energy balances within the solitons.

© 2018 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  29. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
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  31. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013).
    [Crossref]
  32. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
    [Crossref]
  33. J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).
    [Crossref]

2018 (3)

2017 (3)

X. K. Yao and X. M. Liu, “Beam dynamics in disordered PT-symmetric optical lattices based on eigenstate analyses,” Phys. Rev. A 95, 033804 (2017).
[Crossref]

C. Huang and L. Dong, “Beam propagation management in a fractional Schrödinger equation,” Sci. Rep. 7, 5442 (2017).
[Crossref]

X. Huang, X. Shi, Z. Deng, Y. Bai, and X. Fu, “Potential barrier-induced dynamics of finite energy Airy beams in fractional Schrödinger equation,” Opt. Express 25, 32560–32569 (2017).
[Crossref]

2016 (8)

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref]

Q. Li and X. Wu, “Soliton solutions for fractional Schrödinger equations,” Appl. Math. Lett. 53, 119–124 (2016).
[Crossref]

Y. Wan and Z. Wang, “Bound state for fractional Schrödinger equation with saturable nonlinearity,” Appl. Math. Comput. 273, 735–740 (2016).
[Crossref]

C. Huang and L. Dong, “Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice,” Opt. Lett. 41, 5636–5639 (2016).
[Crossref]

L. Zhang, C. Li, H. Zhong, C. Xu, D. Lei, Y. Li, and D. Fan, “Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes,” Opt. Express 24, 14406–14418 (2016).
[Crossref]

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

Y. V. Kartashov, C. Hang, G. Huang, and L. Torner, “Three-dimensional topological solitons in PT-symmetric optical lattices,” Optica 3, 1048–1055 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

2015 (2)

S. Longhi, “Fractional Schrödinger equation in optics,” Opt. Lett. 40, 1117–1120 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (2015).
[Crossref]

2013 (2)

B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88, 012120 (2013).
[Crossref]

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

2012 (2)

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

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

2010 (2)

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

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

2009 (1)

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

2008 (3)

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

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]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).
[Crossref]

2007 (1)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

2002 (1)

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
[Crossref]

2000 (2)

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
[Crossref]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
[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]

Ahmed, N.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref]

Bai, Y.

Belic, M. R.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (2015).
[Crossref]

Bender, C. M.

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

Christodoulides, D.

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

Christodoulides, D. N.

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

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

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

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

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]

Cui, Y. D.

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time dynamics of the build-up of solitons in mode-locked lasers,” Phys. Rev. Lett. 121, 023905 (2018).

Deng, Z.

Dmitriev, S. V.

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

Dong, L.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, 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, 192–195 (2010).
[Crossref]

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

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]

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

Fan, D.

Fu, X.

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

Guo, B.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

Hang, C.

Herrmann, R.

R. Herrmann, Fractional Calculus: An Introduction for Physicists (World Scientific, 2011).

Huang, C.

Huang, D.

B. Guo and D. Huang, “Existence and stability of standing waves for nonlinear fractional Schrödinger equations,” J. Math. Phys. 53, 083702 (2012).
[Crossref]

Huang, G.

Huang, J.

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

Huang, X.

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, 192–195 (2010).
[Crossref]

Kivshar, Y. S.

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

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).
[Crossref]

Laskin, N.

N. Laskin, “Fractional Schrödinger equation,” Phys. Rev. E 66, 056108 (2002).
[Crossref]

N. Laskin, “Fractional quantum mechanics and Lévy path integrals,” Phys. Lett. A 268, 298–305 (2000).
[Crossref]

N. Laskin, “Fractional quantum mechanics,” Phys. Rev. E 62, 3135–3145 (2000).
[Crossref]

Lee, C.

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

Lei, D.

Li, C.

Li, Q.

Q. Li and X. Wu, “Soliton solutions for fractional Schrödinger equations,” Appl. Math. Lett. 53, 119–124 (2016).
[Crossref]

Li, Y.

Liu, X.

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (2015).
[Crossref]

Liu, X. M.

X. M. Liu, X. K. Yao, and Y. D. Cui, “Real-time dynamics of the build-up of solitons in mode-locked lasers,” Phys. Rev. Lett. 121, 023905 (2018).

X. K. Yao and X. M. Liu, “Beam dynamics in disordered PT-symmetric optical lattices based on eigenstate analyses,” Phys. Rev. A 95, 033804 (2017).
[Crossref]

Longhi, S.

Lumer, Y.

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

Makris, K. G.

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

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

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

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]

Morandotti, R.

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

Musslimani, Z. H.

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

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

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

Nixon, S.

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

Plotnik, Y.

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

Rechtsman, M. C.

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

Rüter, C. E.

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

Salamo, G. J.

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

Segev, M.

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

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

Shi, X.

Siviloglou, G.

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

Stickler, B. A.

B. A. Stickler, “Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal,” Phys. Rev. E 88, 012120 (2013).
[Crossref]

Suchkov, S. V.

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

Sukhorukov, A. A.

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

Tian, Z.

Torner, L.

Wan, Y.

Y. Wan and Z. Wang, “Bound state for fractional Schrödinger equation with saturable nonlinearity,” Appl. Math. Comput. 273, 735–740 (2016).
[Crossref]

Wang, Z.

Y. Wan and Z. Wang, “Bound state for fractional Schrödinger equation with saturable nonlinearity,” Appl. Math. Comput. 273, 735–740 (2016).
[Crossref]

Wu, X.

Q. Li and X. Wu, “Soliton solutions for fractional Schrödinger equations,” Appl. Math. Lett. 53, 119–124 (2016).
[Crossref]

Xiao, J.

Xiao, M.

Y. Zhang, H. Zhong, M. R. Belić, N. Ahmed, Y. Zhang, and M. Xiao, “Diffraction-free beams in fractional Schrödinger equation,” Sci. Rep. 6, 23645 (2016).
[Crossref]

Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
[Crossref]

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Y. Zhang, H. Zhong, M. R. Belić, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. 10, 526–531 (2016).
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Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (2015).
[Crossref]

Y. Zhang, X. Liu, M. R. Belić, W. Zhong, Y. Zhang, and M. Xiao, “Propagation dynamics of a light beam in a fractional Schrödinger equation,” Phys. Rev. Lett. 115, 180403 (2015).
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Figures (6)

Fig. 1.
Fig. 1. (a) Photonic band structure for FSE with PT-symmetric periodic potential, when Vi=0.4 (red line) and 0.55 (blue line). (b) Imaginary parts of the band structure for Vi=0.55.
Fig. 2.
Fig. 2. (a) Soliton field profile (real part: blue line, imaginary part: red line) for β=0.8. (b) The spectral distribution |ϕ˜(k)| in the inverse space. (c) Transverse power flow (red line) of the soliton. (d) Spectrum of the linearization operator for the soliton solution in (a). (e) Stable propagation perturbed with weak noise. The light gray line in (a) and (d) represents the scaled real part of the potential. For all cases, Vi=0.4 and A=1.
Fig. 3.
Fig. 3. (a) Soliton field profile (real part: blue line, imaginary part: red line). The light gray line denotes the real part of the potential. (b) The perturbation growth rate profile for the soliton solution in (a). (c) Unstable PT soliton propagation perturbed with weak noise. In all cases, Vi=0.55 and β=0.8.
Fig. 4.
Fig. 4. (a) Domains of stability on the plane (β, Vi) for 1D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) The maximum real part of perturbation growth rate versus β for Vi=0.4 and A=1.
Fig. 5.
Fig. 5. (a) Band structure of a 2D-PT potential for Eq. (4) in linear form. The inset displays the band structure in the reduced Brillouin zone. (b) The profile of a 2D soliton when β=5.55. (c) The spectral profile |ϕ˜(kx,ky)| in the inverse space. (d) Transverse power flow (indicated by arrows) for the soliton in (b) within one cell. “L” and “G” indicate the loss and gain regions, respectively. For all cases, A=4 and Vi=0.2.
Fig. 6.
Fig. 6. (a) Domains of stability on the plane (β, Vi) for 2D solitons. Solitons are stable in the blue region. The red line denotes the lower edge of the semi-infinite forbidden gap. (b) Real part of perturbation growth rate versus β for A=4 and Vi=0.2.

Equations (4)

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iψz(2x2)α/2ψ+V(x)ψ+|ψ|2ψ=0,
nAmnBn|k+Km|αAm=βAm,
i(Lϕ2ϕ*2L*)(vw)=λ(vw),
iψz(2x22y2)α/2ψ+V(x,y)ψ+|ψ|2ψ=0.

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