Abstract

We report the existence and stability properties of multipeaked solitons in a defocusing Kerr medium with an imprinted complex optical lattice featuring a parity-time (PT) symmetry. Various families of soliton solutions with a different number of peaks are found in the first finite gap of the lattice. Linear stability analysis corroborated by direct propagation simulations reveals that multipeaked gap solitons can propagate stably in a wide range, provided that their propagation constant exceeds a critical value. Our findings demonstrate, for the first time, the existence of stable multipeaked gap solitons in a PT-symmetric lattice.

© 2012 Optical Society of America

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  1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
    [CrossRef]
  3. C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
    [CrossRef]
  4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
  5. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
    [CrossRef]
  6. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
    [CrossRef]
  7. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
    [CrossRef]
  8. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  9. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef]
  10. K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010).
    [CrossRef]
  11. H. Wang and J. Wang, Opt. Express 19, 4030 (2011).
    [CrossRef]
  12. Z. Lu and Z.-M. Zhang, Opt. Express 19, 11457 (2011).
    [CrossRef]
  13. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
    [CrossRef]
  14. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
    [CrossRef]
  15. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
    [CrossRef]
  16. Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
    [CrossRef]
  17. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
    [CrossRef]
  18. T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
    [CrossRef]
  19. Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007).
    [CrossRef]
  20. Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
    [CrossRef]
  21. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
    [CrossRef]
  22. S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
    [CrossRef]
  23. M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
    [CrossRef]

2012 (2)

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

2011 (6)

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

H. Wang and J. Wang, Opt. Express 19, 4030 (2011).
[CrossRef]

Z. Lu and Z.-M. Zhang, Opt. Express 19, 11457 (2011).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

2010 (2)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. Zhou, Z. Guo, J. Wang, and S. Liu, Opt. Lett. 35, 2928 (2010).
[CrossRef]

2009 (3)

Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
[CrossRef]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

2007 (1)

Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007).
[CrossRef]

2006 (1)

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
[CrossRef]

2005 (2)

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
[CrossRef]

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

1999 (2)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
[CrossRef]

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

1998 (1)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Abdullaev, F. K.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, 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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Alexander, T. J.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
[CrossRef]

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
[CrossRef]

Bender, C. M.

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Boettcher, S.

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Christodoulides, D.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

Christodoulides, D. N.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Darmanyan, S.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Dunne, G. V.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
[CrossRef]

El-Ganainy, R.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Ge, L.

S. Nixon, L. Ge, and J. Yang, 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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Guo, Z.

He, Y.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Jiang, X.

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

Jones, H. F.

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Kartashov, Y. V.

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Y. S.

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
[CrossRef]

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
[CrossRef]

Kobyakov, A.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Konotop, V. V.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Lederer, F.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Li, H.

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

Liang, Z.

Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
[CrossRef]

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Liu, S.

Lu, Z.

Makris, K.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

Meisinger, P. N.

C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
[CrossRef]

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Muga, J. G.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Musslimani, Z.

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
[CrossRef]

Nixon, S.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Ostrovskaya, E. A.

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
[CrossRef]

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Shi, Z.

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007).
[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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Vázquez, L.

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Wang, H.

Wang, J.

Wu, B.

Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
[CrossRef]

Yang, J.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
[CrossRef]

Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007).
[CrossRef]

Zezyulin, D. A.

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
[CrossRef]

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Zhang, Y.

Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
[CrossRef]

Zhang, Z.-M.

Zhou, K.

Zhu, X.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

Europhys. Lett. (1)

D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, Europhys. Lett. 96, 64003 (2011).
[CrossRef]

Int. J. Theor. Phys. (1)

K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Int. J. Theor. Phys. 50, 1019 (2011).
[CrossRef]

J. Phys. A (1)

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Lett. A (1)

C. M. Bender, G. V. Dunne, and P. N. Meisinger, Phys. Lett. A 252, 272 (1999).
[CrossRef]

Phys. Rev. A (6)

F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
[CrossRef]

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Z. Shi, X. Jiang, X. Zhu, and H. Li, Phys. Rev. A 84, 053855 (2011).
[CrossRef]

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Y. Zhang, Z. Liang, and B. Wu, Phys. Rev. A 80, 063815 (2009).
[CrossRef]

J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
[CrossRef]

Phys. Rev. B (1)

S. Darmanyan, A. Kobyakov, F. Lederer, and L. Vázquez, Phys. Rev. B 59, 5994 (1999).
[CrossRef]

Phys. Rev. E (1)

Z. Shi and J. Yang, Phys. Rev. E 75, 056602 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

T. J. Alexander, E. A. Ostrovskaya, and Y. S. Kivshar, Phys. Rev. Lett. 96, 040401 (2006).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, 270401 (2002).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Nonlinear Bloch waves originate from the lower edge of the first band. (a) Amplitude (|ϕ|max) versus propagation constant b. (b), (c) Profiles and moduli of nonlinear Bloch waves marked in (a) at b=2.2 and 4.0, respectively. To distinguish with ϕr, |ϕ| is enhanced by 0.5.

Fig. 2.
Fig. 2.

Profiles and moduli of (a) three-, (b) five-, (c) seven-, and (d) nine-peak solitons marked in Fig. 3(a) at b=4.0. Solid: real part; Dashed: imaginary part. For comparison, |ϕ| of the corresponding nonlinear Bloch wave is shown in (d) [see also Fig. 1(c)]. To distinguish with ϕr, |ϕ| is enhanced by 0.5.

Fig. 3.
Fig. 3.

(a) Power P versus b for solitons with a different number of peaks. (b) Power and amplitude (inset) ratios between the imaginary and real parts of solitons. (c) Power flow of seven-peak soliton at b=4.0.

Fig. 4.
Fig. 4.

(a), (b) Spectra of the linearization operator for nine- and three-peak solitons at b=4.0 and 2.7, respectively. (c), (d) Stable and unstable propagations of solitons with eigenvalue shown in (a) and (b).

Equations (3)

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iψz+2ψx2[V(x)+iW(x)]ψ|ψ|2ψ=0.
d2ϕdx2+bϕ[V(x)+iW(x)]ϕ|ϕ|2ϕ=0,
i[iW+iIm(ϕ2)L^+Re(ϕ2)L^Re(ϕ2)iWiIm(ϕ2)][vw]=λ[vw],

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