Abstract

We numerically study the gray solitons in parity-time (PT) symmetric potentials. Simulated results show that there are two kinds of gray solitons, the dip-shaped gray solitons and the hump-shaped solitons, and both of them can be stable. Hump-shaped solitons can always exist, but the grayness of a stable dip-shaped gray soliton should exceed a threshold value. More interesting, it is discovered that when propagating in PT symmetric potentials, the gray solitons have no transverse deviation, and this is a phenomenon different from the usual gray solitons.

© 2011 Optical Society of America

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  1. G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
    [CrossRef] [PubMed]
  2. Y. V. Kartashov and L. Torner, Opt. Lett. 32, 946 (2007).
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    [CrossRef]
  4. G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
    [CrossRef] [PubMed]
  5. B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett. 19, 1816 (1994).
    [CrossRef] [PubMed]
  6. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81(1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  9. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192(2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Lett. 30, 3171(2005).
    [CrossRef] [PubMed]
  13. M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140(2005).
    [CrossRef] [PubMed]
  14. Y. S. Kivshar and W. Królikowski, Opt. Lett. 20, 1527 (1995).
    [CrossRef] [PubMed]
  15. D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).
    [CrossRef]

2010 (2)

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

Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, and L. Torner, Opt. Lett. 35, 1638 (2010).
[CrossRef] [PubMed]

2008 (3)

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, 244019 (2008).
[CrossRef]

2007 (1)

2005 (3)

1998 (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81(1998).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

1992 (1)

G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

1991 (1)

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Ablowitz, M. J.

Andersen, D. R.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Christodoulides, D. N.

C. E. Ruter, 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] [PubMed]

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, 244019 (2008).
[CrossRef]

El-Ganainy, R.

C. E. Ruter, 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, J. Phys. A 41, 244019 (2008).
[CrossRef]

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

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

Frantzeskakis, D. J.

D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Kartashov, Y. V.

Kevrekidis, P. G.

D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).
[CrossRef]

Kip, D.

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

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81(1998).
[CrossRef]

Y. S. Kivshar and W. Królikowski, Opt. Lett. 20, 1527 (1995).
[CrossRef] [PubMed]

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

Konotop, V. V.

Królikowski, W.

Law, C. T.

G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

Luther-Davies, B.

Makris, K. G.

C. E. Ruter, 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] [PubMed]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, 244019 (2008).
[CrossRef]

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

Musslimani, Z. H.

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

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

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, 244019 (2008).
[CrossRef]

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

Pelinovsky, D. E.

D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).
[CrossRef]

Powles, R.

Regan, J. J.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Ruter, C. E.

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

Segev, M.

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

Swartzlander, G. A.

G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

Tikhonenko, V.

Torner, L.

Vysloukh, V. A.

Xu, Z.

Yin, H.

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

J. Phys. A (1)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, J. Phys. A 41, 244019 (2008).
[CrossRef]

Nat. Phys. (1)

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

Opt. Lett. (6)

Phys. Rep. (1)

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81(1998).
[CrossRef]

Phys. Rev. E (1)

D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis, Phys. Rev. E 72, 016615 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

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

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

G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) Real (dashed black curve) and imaginary (solid pink curve) components of the dip-shaped gray soliton; (b) I (solid pink curve) and ϕ (dashed black curve) of the soliton at b = 1.46 . The dash-dotted blue and dotted red curves in (a) and (b) correspond to the functions of V ( x ) and W ( x ) , respectively. (c) Error of numerical solutions versus b, points marked with circles correspond to the cases shown in (f) and (g). (d) Q versus g d , point marked with circle corresponds to the case shown in (g). (e) Energy flow density of the soliton at b = 1.46 . (f)–(j) Simulated propagation of the solitons with 1% random noise at b = 0.78 , 0.80 , 1.00 , 2.34 , and 5.00 , respectively.

Fig. 2
Fig. 2

(a) Real (dashed black curve) and imaginary (solid pink curve) components of the hump-shaped gray soliton; (b) I (solid pink curve) and ϕ (dashed black curve) of the soliton at b = 0.74 . The dash-dotted blue and dotted red curves in (a) and (b) correspond to the functions of V ( x ) and W ( x ) , respectively. (c) Error of numerical solutions versus b. (d)  Im ( δ ) versus b. (e) Energy flow density of the soliton at b = 0.74 . (f)–(j) Simulated propagation of the solitons with 1% random noise at b = 0.05 , 0.74 , 1.49 , 2.48 , and 4.73 , respectively.

Fig. 3
Fig. 3

(a) Error of numerical dip-shaped gray solutions versus W 0 ; points marked with circles correspond to cases shown in (c) and (d). (b) Q versus g d of the dip-shaped gray solitons; point marked with circle corresponds to the case shown in (d). (c) and (d) Simulated propagation of the gray soliton with 1% noise at W 0 = 1.37 and 1.34, respectively. (e) Errors of numerical hump-shaped solutions versus W 0 ; points marked with circles correspond to the cases shown in (g) and (h). (f)  Im ( δ ) versus b of the hump-shaped gray solitons; point marked with circle corresponds to the case shown in (h). (g)–(i) Simulated propagation of the hump-shaped gray solitons with 1% noise at W 0 = 0.58 , 0.56, and 0.48, respectively.

Equations (4)

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i q z + 2 q x 2 + [ V ( x ) + i W ( x ) ] q | q | 2 q = 0 ,
2 u x 2 + [ V ( x ) + i W ( x ) ] u | u | 2 u b u = 0 .
δ F = ( 2 x 2 + V ( x ) + i W ( x ) 2 | u | 2 b ) F + u 2 G ,
δ G = ( 2 x 2 V ( x ) + i W ( x ) + 2 | u | 2 + b ) G u * 2 F .

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