Abstract

We study the solitons in parity-time symmetric potential in the medium with spatially modulated nonlocal nonlinearity. It is found that the coefficient of the spatially modulated nonlinearity and the degree of the uniform nonlocality can profoundly affect the stability of solitons. There exist stable solitons in low-power region, and unstable solitons in high-power region. In the unstable cases, the solitons exhibit jump from the original site to the next one, and they can continue the motion into the other lattices. The region of the stable soliton can be expanded by increasing the coefficient of the modulated nonlocality. Finally, critical amplitude of the imaginary part of the linear PT lattices is obtained, above which solitons are unstable and decay immediately.

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  1. C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
    [Crossref]
  2. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
    [Crossref] [PubMed]
  3. K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).
  4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).
  5. C. E. Rüter, K. G. Makris, R. Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
  6. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
    [Crossref] [PubMed]
  7. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
    [Crossref] [PubMed]
  8. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
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    [Crossref] [PubMed]
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    [Crossref]
  12. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
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    [Crossref]
  14. F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-Symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
    [Crossref]
  15. D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
    [Crossref]
  16. X. Zhu, H. Wang, L. X. Zheng, H. G. Li, and Y. J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
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  17. H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
    [Crossref] [PubMed]
  18. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
    [Crossref]
  19. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
    [Crossref]
  20. 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(15), 3320–3324 (2012).
    [Crossref]
  21. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
    [Crossref] [PubMed]
  22. F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
    [Crossref]
  23. Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
    [Crossref] [PubMed]
  24. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
    [Crossref]
  25. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
    [Crossref] [PubMed]
  26. Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
    [Crossref] [PubMed]
  27. D. Blömer, A. Szameit, F. Dreisow, T. Schreiber, S. Nolte, and A. Tünnermann, “Nonlinear refractive index of fs-laser-written waveguides in fused silica,” Opt. Express 14(6), 2151–2157 (2006).
    [Crossref] [PubMed]
  28. M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
    [Crossref] [PubMed]
  29. G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
    [Crossref]
  30. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
    [Crossref]
  31. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
    [Crossref] [PubMed]
  32. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
    [Crossref]
  33. M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
    [Crossref] [PubMed]

2012 (8)

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 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(15), 3320–3324 (2012).
[Crossref]

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

2011 (7)

2010 (5)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

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

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

2008 (4)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

2006 (2)

2005 (3)

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

2004 (1)

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

1998 (1)

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Abdullaev, F. Kh.

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-Symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[Crossref]

Ablowitz, M. J.

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Akylas, T. I.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

Bender, C. M.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Blömer, D.

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Chen, D.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Chen, L.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

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(15), 3320–3324 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Christodoulides, D. N.

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Dreisow, F.

Driben, R.

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

EI-Ganainy, R.

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Ganainy, R.

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

Ge, L.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

Grimm, R.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Guo, Z.

He, Y.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 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(15), 3320–3324 (2012).
[Crossref]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

He, Y. J.

Hecker Denschlag, J.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Hellwig, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Hu, B.

Hu, S.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Hu, W.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Hwang, G.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

Jiang, X.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Kartashov, Y. V.

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-Symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[Crossref]

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Kip, D.

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

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (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(4), 041805 (2011).
[Crossref]

Kottos, T.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Li, H.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Li, H. G.

Li, L.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Li, R.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

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(15), 3320–3324 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Liu, S.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
[Crossref] [PubMed]

Lu, D.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Lu, K.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Ma, C.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Ma, X.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[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(6), 063807 (2010).

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Malomed, B. A.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
[Crossref] [PubMed]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Mihalache, D.

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(15), 3320–3324 (2012).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

Nixon, S.

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

Nolte, S.

Ruff, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Rüter, C. E.

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

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Schreiber, T.

Segev, M.

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

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

Shi, Z.

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Szameit, A.

Thalhammer, G.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Theis, M.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Torner, L.

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Tünnermann, A.

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

Vysloukh, V. A.

Wang, H.

Wang, J.

Winkler, K.

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Xu, Z.

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

Yang, J.

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

Yang, N.

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Ye, F.

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (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(4), 041805 (2011).
[Crossref]

Zhang, Y.

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (2012).
[Crossref]

Zheng, L. X.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

Zhou, K.

Zhu, X.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 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(15), 3320–3324 (2012).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

X. Zhu, H. Wang, L. X. Zheng, H. G. Li, and Y. J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
[Crossref] [PubMed]

Nat. Phys. (1)

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

Opt. Commun. (2)

S. Liu, C. Ma, Y. Zhang, and K. Lu, “Bragg gap solitons in PT symmetric lattices with competing nonlinearity,” Opt. Commun. 285(7), 1934–1939 (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(15), 3320–3324 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (8)

X. Zhu, H. Wang, L. X. Zheng, H. G. Li, and Y. J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36(14), 2680–2682 (2011).
[Crossref] [PubMed]

H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in parity-time symmetric potentials,” Opt. Lett. 36(16), 3290–3292 (2011).
[Crossref] [PubMed]

R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36(22), 4323–4325 (2011).
[Crossref] [PubMed]

Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31(17), 2595–2597 (2006).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008).
[Crossref] [PubMed]

Y. He, D. Mihalache, and B. Hu, “Soliton drift, rebound, penetration, and trapping at the interface between media with uniform and spatially modulated nonlinearities,” Opt. Lett. 35(10), 1716–1718 (2010).
[Crossref] [PubMed]

K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35(17), 2928–2930 (2010).
[Crossref] [PubMed]

M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30(16), 2140–2142 (2005).
[Crossref] [PubMed]

Phys. Rev. A (8)

S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85(2), 023822 (2012).
[Crossref]

F. Ye, Y. V. Kartashov, and L. Torner, “Nonlocal surface dipoles and vortices,” Phys. Rev. A 77(3), 033829 (2008).
[Crossref]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, “Defect solitons in parity-time-symmetric optical lattices with nonlocal nonlinearity,” Phys. Rev. A 85(4), 043826 (2012).
[Crossref]

H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010).
[Crossref]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81(6), 063807 (2010).

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in PT-Symmetric nonlinear lattices,” Phys. Rev. A 83(4), 041805 (2011).
[Crossref]

Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defousing Kerr media with PT-symmetric potentials,” Phys. Rev. A 84(5), 053855 (2011).
[Crossref]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattices solitons in PT-Symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85(1), 013831 (2012).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005).
[Crossref] [PubMed]

Phys. Rev. Lett. (8)

M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, “Tuning the Scattering Length with an Optically Induced Feshbach Resonance,” Phys. Rev. Lett. 93(12), 123001 (2004).
[Crossref] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95(11), 113901 (2005).
[Crossref] [PubMed]

D. A. Zezyulin and V. V. Konotop, “Nonlinear Modes in Finite-Dimensional PT -Symmetric Systems,” Phys. Rev. Lett. 108(21), 213906 (2012).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry Breaking in Complex Optical Potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref] [PubMed]

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially Fragile PT Symmetry in Lattices with Localized Eigenmodes,” Phys. Rev. Lett. 103(3), 030402 (2009).
[Crossref] [PubMed]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical Solitons in PT Periodic Potentials,” Phys. Rev. Lett. 100(3), 030402 (2008).
[Crossref] [PubMed]

K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam Dynanics in PT Symmetric Optical Lattices,” Phys. Rev. Lett. 100, 103904 (2008).

Proc. Romanian Acad. A (1)

L. Chen, R. Li, N. Yang, D. Chen, and L. Li, “Optical modes in PT-symmetric double-channel waveguides,” Proc. Romanian Acad. A 13, 46–54 (2012).

Rev. Mod. Phys. (1)

Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011).
[Crossref]

Stud. Appl. Math. (1)

G. Hwang, T. I. Akylas, and J. Yang, “Solitary Waves and Their Linear Stability in Nonlinear Lattices,” Stud. Appl. Math. 128(3), 275–298 (2012).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

(a) PT-symmetric optical lattices (OLs) with W0 = 2 (the bue line is the real part, and the red line is the imaginary part). (b)The band structure corresponding to the lattice profile shown in panel (a).

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2

Power P versus propagation constant μ for various k with d = 1 and W0 = 2.The solid line represents the stable regions and the dash line represents the unstable regions.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

Soliton profiles (blue line is the real part, the red line is the imaginary part, and the green dash line is the refractive index profile) and Soliton evolution for μ = 1.8 (a, c), and μ = 2.5 (b, d). The other parameters are d = 1, W0 = 2, and k = −0.8.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

The soliton profiles for μ = 2.0 (a), and μ = 2.8 (b). (c) and (d) are the corresponding propagations. Here, k = 0. The other parameters are the same as those in Fig. 3.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5

The soliton profiles for μ = 2.2 (a), and μ = 3.0 (b). (c), (d) are the corresponding propagations. Here, k = 1. The other parameters are the same as those in Fig. 3.

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6

When μ = 2.0. (a). (b), (c) are the soliton profiles for k = −0.8, k = 0, and k = 1, respectively.

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7

Power P versus propagation constant μ (a). The soliton profiles for μ = 1.9 (b), and μ = 2.5 (c). (d), (e) are the corresponding propagations. Here, d = 3. The other parameters are the same as those in Fig. 5.

Download Full Size | PPT Slide | PDF

Fig. 8
Fig. 8

Stability domain (μ, k) for (a) W0 = 1.2, (b) W0 = 2, and (c) W0 = 2.4 when d = 1. (d) stability domain (μ, d) for W0 = 2 and k = 1. The gray regions are stable domains.

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Fig. 9
Fig. 9

(a) PT symmetric OLs with W0 = 4.2 (the blue line is the real part, and the red line is the imaginary part). (b) The band structure corresponding to the lattice profile shown in panel (a).

Download Full Size | PPT Slide | PDF

Fig. 10
Fig. 10

The soliton profiles for μ = 1.0 (a) and μ = 2.0 (b), and. (c) and (d) are their corresponding propagations. The parameters are d = 1, W0 = 4.2, and k = 1.

Download Full Size | PPT Slide | PDF

Equations (5)

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

i ∂q ∂z + 1 2 ∂ 2 q ∂ x 2 +(V+iW)q+[1+f(x)]q ∫ −∞ +∞ g(x−λ) | q(λ) | 2 dλ=0,
g(x)=1/(2 d 1 2 )exp(−| x |/ d 1 2 )
1 2 ∂ 2 u ∂ x 2 +[V(x)+iW(x)]u+[1+f(x)]u ∫ −∞ +∞ g(x−λ) | u(λ) | 2 dλ−μu=0.
1 2 ∂ 2 u ∂ x 2 +[V(x)+iW(x)]u=μu.
1 2 ( ∂ 2 ∂ x 2 +2iK ∂ ∂x − K 2 ) F K +[V(x)+iW(x)] F K =μ F K .

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