Abstract

We report on the existence and stability of nonlocal multihump gap solitons in one-dimensional parity-time symmetric periodic potentials. They can exist in the first gap in defocusing nonlocal nonlinearity and in the semi-infinite gap in focusing nonlocal nonlinearity. These solitons can be stable in the defocusing nonlinearity but are unstable in the focusing nonlinearity. For the multihump solitons, the shapes of the nonlinear contribution to refractive index are also multihump. The stability and shapes of the intensity distribution of these solitons will be changed by the degree of nonlocality. We also study the transverse power flow of these solitons.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998).
    [CrossRef]
  2. Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef]
  3. K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
  4. K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef]
  6. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012).
    [CrossRef]
  7. 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, 093902 (2009).
    [CrossRef]
  8. C. E. Rüter, K. G. Makris, R. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
    [CrossRef]
  9. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
    [CrossRef]
  10. X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, “Gap solitons in parity-time complex periodic optical lattices with the real part of superlattices,” Opt. Lett. 36, 2680–2682 (2011).
    [CrossRef]
  11. H. Li, Z. Shi, X. Jiang, and X. Zhu, “Gray solitons in PT symmetric potentials,” Opt. Lett. 36, 3290–3292 (2011).
    [CrossRef]
  12. K. Zhou, Z. Guo, J. Wang, and S. Liu, “Defect modes in defective parity-time symmetric periodic complex potentials,” Opt. Lett. 35, 2928–2930 (2010).
    [CrossRef]
  13. H. Wang and J. Wang, “Defect solitons in parity-time periodic potentials,” Opt. Express 19, 4030–4035 (2011).
    [CrossRef]
  14. 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, 043826 (2012).
    [CrossRef]
  15. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012).
    [CrossRef]
  16. 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]
  17. 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]
  18. 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, 013831 (2012).
    [CrossRef]
  19. R. Driben and B. A. Malomed, “Stability of solitons in parity-time-symmetric couplers,” Opt. Lett. 36, 4323–4325 (2011).
    [CrossRef]
  20. H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
    [CrossRef]
  21. Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
    [CrossRef]
  22. C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20, 16823–16831 (2012).
    [CrossRef]
  23. C. Li, C. Huang, H. Liu, and L. Dong, “Multi-peaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (2012).
    [CrossRef]
  24. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
    [CrossRef]
  25. C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312–3314 (2006).
    [CrossRef]
  26. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
    [CrossRef]
  27. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef]
  28. Z. Xu, Y. V. Kartashov, and L. Torner, “Soliton mobility in nonlocal optical lattices,” Phys. Rev. Lett. 95, 113901 (2005).
    [CrossRef]
  29. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Lattice-supported surface solitons in nonlocal nonlinear media,” Opt. Lett. 31, 2595–2597 (2006).
    [CrossRef]
  30. F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35, 628–630 (2010).
    [CrossRef]
  31. Y. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
    [CrossRef]
  32. 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]
  33. M. J. Ablowitz and Z. H. Musslimani, “Spectral renormalization method for computing self-localized solutions to nonlinear systems,” Opt. Lett. 30, 2140–2142 (2005).
    [CrossRef]
  34. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
    [CrossRef]

2013

Y. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

2012

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

C. Li, H. Liu, and L. Dong, “Multi-stable solitons in PT-symmetric optical lattices,” Opt. Express 20, 16823–16831 (2012).
[CrossRef]

C. Li, C. Huang, H. Liu, and L. Dong, “Multi-peaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (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, 013831 (2012).
[CrossRef]

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

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (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, 043826 (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]

2011

2010

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35, 628–630 (2010).
[CrossRef]

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

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

2009

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[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, 093902 (2009).
[CrossRef]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
[CrossRef]

2008

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

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

2007

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]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef]

2006

2005

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

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

2002

1998

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]

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, 093902 (2009).
[CrossRef]

Alexander, T. J.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
[CrossRef]

Alfassi, B.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

Assanto, G.

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]

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[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]

Brzdakiewicz, K. A.

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef]

Chen, Z.

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, 013831 (2012).
[CrossRef]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

K. G. Makris, R. EI-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. EI-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010).
[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, 093902 (2009).
[CrossRef]

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

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

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

Cohen, O.

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef]

Dong, L.

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, 093902 (2009).
[CrossRef]

EI-Ganainy, R.

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

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

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

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[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]

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, 093902 (2009).
[CrossRef]

Guo, Z.

He, Y.

Y. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[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, 013831 (2012).
[CrossRef]

He, Y.-J.

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, 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, 043826 (2012).
[CrossRef]

Huang, C.

Jiang, X.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

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

Kartashov, Y. V.

Kip, D.

C. E. Rüter, K. G. Makris, R. EI-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.

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]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef]

Konotop, V. V.

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

Kottos, T.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[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]

Li, C.

Li, H.

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

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

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

Liu, H.

Liu, J.

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, 013831 (2012).
[CrossRef]

Liu, S.

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, 043826 (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, 043826 (2012).
[CrossRef]

Makris, K. G.

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

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

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

Malomed, B. A.

Y. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

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

Manela, O.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

Mihalache, D.

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, 013831 (2012).
[CrossRef]

Miri, M. A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (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. 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, 093902 (2009).
[CrossRef]

Musslimani, Z. H.

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

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

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

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

Peccianti, M.

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

Rotschild, C.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312–3314 (2006).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. EI-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, 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, 093902 (2009).
[CrossRef]

Segev, M.

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

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312–3314 (2006).
[CrossRef]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Shi, Z.

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

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

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, 093902 (2009).
[CrossRef]

Torner, L.

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, 093902 (2009).
[CrossRef]

Vysloukh, V. A.

Wang, H.

Wang, J.

Xu, Z.

Yang, J.

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

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
[CrossRef]

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]

Ye, F.

Zezyulin, D. A.

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

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, 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, 013831 (2012).
[CrossRef]

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (2012).
[CrossRef]

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

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

Europhys. Lett.

Z. Shi, H. Li, X. Zhu, and X. Jiang, “Nonlocal bright spatial solitons in defousing Kerr media supported by PT-symmetric potentials,” Europhys. Lett. 98, 64006 (2012).
[CrossRef]

Nat. Phys.

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

Nature

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

C. Li, C. Huang, H. Liu, and L. Dong, “Multi-peaked gap solitons in PT-symmetric optical lattices,” Opt. Lett. 37, 4543–4545 (2012).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002).
[CrossRef]

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312–3314 (2006).
[CrossRef]

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

F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Twin-vortex solitons in nonlocal nonlinear media,” Opt. Lett. 35, 628–630 (2010).
[CrossRef]

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

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

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

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

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

Phys. Rev. A

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

H. Li, X. Jiang, X. Zhu, and Z. Shi, “Nonlocal solitons in dual-periodic PT-symmetric optical lattices,” Phys. Rev. A 86, 023840 (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, 043826 (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]

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]

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]

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, 013831 (2012).
[CrossRef]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, “Truncated-Bloch-wave solitons in optical latteces,” Phys. Rev. A 79, 043610 (2009).
[CrossRef]

Y. He and B. A. Malomed, “Solitary modes in nonlocal media with inhomogeneous self-repulsive nonlinearity,” Phys. Rev. A 87, 053812 (2013).
[CrossRef]

Phys. Rev. Lett.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, “Nonlocal surface-wave solitons,” Phys. Rev. Lett. 98, 213901 (2007).
[CrossRef]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite solitons clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef]

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

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

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

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

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, “Exponentially fragile PT symmetry in lattices with localized eigenmodes,” Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 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, 093902 (2009).
[CrossRef]

Stud. Appl. Math.

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]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1.
Fig. 1.

(a) is the band structure. (b) and (c) are the power diagrams (the shaded regions are the Bloch bands, the solid lines represent stable cases, and the dashed lines represent unstable cases) for one-hump, three-hump, and seven-hump solitons when d=0.5 and d=3, respectively. (d), (e), and (f) show the soliton profile (the solid line is the real part, and the dashed line is the imaginary part), the distribution of refractive index, and the transverse power flow of the soliton for σ=1, μ=3.35, and d=0.5, respectively. (g), (h), and (i) are the soliton profile, the shape of refractive index distribution, and the transverse power flow of the soliton for σ=1, μ=3.35 and d=3, respectively. V0=10 and W0=0.1.

Fig. 2.
Fig. 2.

(a) and (c) show the amplitude distributions of the three-hump solitons when d=0.5 and d=3, respectively. (b) and (d) are their phase structures. σ=1 and μ=3.35.

Fig. 3.
Fig. 3.

(a) and (d) are the intensity distributions of the three-hump solitons for d=0.5 and d=3, respectively. (b) and (e) are the linear stability spectra for the two solitons. (c) and (f) are the corresponding stable propagation of the two perturbed solitons. σ=1 and μ=3.35.

Fig. 4.
Fig. 4.

(a), (b), and (c) show the distribution of intensity of the three-hump soliton, the linear stability spectrum, and the stable propagation of the perturbed soliton for d=0.5, respectively. (d), (e), and (f) are the shape of intensity distribution of the three-hump soliton, the linear stability spectrum, and the unstable propagation of the perturbed soliton for d=3, respectively. σ=1 and μ=4.3.

Fig. 5.
Fig. 5.

Profiles of solitons (a) and (d), the intensity distributions (b) and (e), and the propagations of the two perturbed solitons for d=0.5 (c) and d=3 (f), respectively. σ=1, μ=4.05, and W0=0.2.

Fig. 6.
Fig. 6.

(a) Profile of the three-hump soliton. (b) Distribution of intensity. (c) Linear stability spectrum. (d) Unstable propagation of the perturbed soliton. σ=1, μ=7.0, and d=0.5.

Fig. 7.
Fig. 7.

(a), (b), and (c)  are the shapes of intensity distribution of the seven-hump soliton, the linear stability spectrum, and the stable propagation of the perturbed soliton for d=0.5, respectively. (d), (e), and (f) show the distribution of intensity of the seven-hump soliton, the linear stability spectrum, and the stable propagation of the perturbed soliton for d=3, respectively. σ=1 and μ=3.35.

Fig. 8.
Fig. 8.

(a), (b), and (c) are the seven-hump soliton profile, soliton intensity distribution, and stable propagation of the perturbed soliton for d=0.5, respectively. The seven-hump soliton profile, soliton intensity distribution, and unstable propagation of the perturbed soliton for d=3 are displayed in (d), (e), and (f), respectively. σ=1 and μ=4.3.

Fig. 9.
Fig. 9.

(a), (b), (c), and (d) are the soliton profile, transverse power flow of soliton, refractive index, and stable propagation of the perturbed soliton for the one hump soliton when μ=3.35 and d=0.5, respectively. (e) The power diagram of three-hump solitons in real optical lattices (W0=0) when d=3. (f), (g), (h), and (i) are, respectively, the soliton profile, zero transverse power flow of soliton, refractive index, and the stable propagation of the perturbed soliton for the three-hump soliton when μ=7.1, W0=0, and d=3. σ=1.

Fig. 10.
Fig. 10.

(a) and (d) are the profiles of the out-of-phase three-hump and four-hump (the peaks are not on the consecutive lattice sites) solitons. (b) and (e) are the intensity distributions of the two solitons. (c) and (f) are the corresponding unstable propagations of the perturbed solitons. σ=1, d=0.5, and μ=3.35.

Fig. 11.
Fig. 11.

(a) and (b) show the propagations of the perturbed three-hump and seven-hump solitons with k=0.1. The propagations of the perturbed three-hump and seven-hump solitons with k=0.5 are shown in (c) and (d), respectively. σ=1, d=0.5, and μ=3.35.

Fig. 12.
Fig. 12.

(a) shows the rectangular PT-symmetric optical lattices for V01=10 and W01=0.02 (the solid and dashed lines are the real and imaginary parts, respectively). (b) and (d) are the profiles of three-hump and seven-hump solitons for d=0.5, μ=3.35, and σ=1. (c) and (e) are the corresponding stable propagation of the two perturbed solitons. (f) and (h) show the profiles of three-hump and seven-hump solitons for d=3, μ=3.35, and σ=1. (g) and (i) are the stable propagations of the two perturbed solitons.

Fig. 13.
Fig. 13.

(a) and (d) are profiles of soliton intensity distributions; (b) and (e) are transverse power flow of the solitons; (c) and (f) are linear stability spectra; (g) and (h) are the unstable propagations of the perturbed solitons for d=0.5 and d=3, respectively. σ=1 and μ=2.5.

Equations (10)

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

iUz+2Ux2+(V+iW)U+σnU=0,
d2nx2n+|U|2=0.
2qx2+(V+iW)q+σnqμq=0,
d2nx2n+|q|2=0.
2hx2+VhWe+σnhμh=0,
2ex2+Ve+Wh+σneμe=0,
d2nx2n+h2+e2=0.
U(x,z)=eiμz[q(x)+F(x)eδz+G*(x)eδ*z],
δF=i[μF+2Fx2+(V+iW)F+σnF+σqΔn],
δG=i[μG2Gx2(ViW)GσnGσq*Δn].

Metrics