Abstract

We report on the existence and stability of the two-dimensional multipeak gap solitons in a parity-time-symmetric periodic potential with defocusing Kerr nonlinearity. We investigate the multipeak solitons with all the peaks of the real parts in-phase. It is found that these solitons can be stable in the first gap. The system can support not only the stable solitons with even number peaks, but also the stable solitons with odd number peaks. The transverse energy flow vector of these solitons is also studied.

© 2013 Optical Society of America

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  1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. Z. H. Musslimani, K. G. Makris, R. EI-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef]
  3. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
    [CrossRef]
  4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  5. O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
    [CrossRef]
  6. D. A. Zezyulin and V. V. Konotop, 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, Phys. Rev. Lett. 103, 093902 (2009).
    [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. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
    [CrossRef]
  10. X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, Opt. Lett. 36, 2680 (2011).
    [CrossRef]
  11. H. Wang and J. Wang, Opt. Express 19, 4030 (2011).
    [CrossRef]
  12. S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
    [CrossRef]
  13. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805 (2011).
    [CrossRef]
  14. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Phys. Rev. A 85, 013831 (2012).
    [CrossRef]
  15. S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
    [CrossRef]
  16. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
    [CrossRef]
  17. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85023822 (2012).
    [CrossRef]
  18. J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
    [CrossRef]
  19. M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140 (2005).
    [CrossRef]
  20. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).
  21. J. Wang, J. Yang, T. J. Alexander, and Y. S. Kivshar, Phys. Rev. A 79, 043610 (2009).
    [CrossRef]
  22. C. Li, C. Huang, H. Liu, and L. Dong, Opt. Lett. 37, 4543 (2012).
    [CrossRef]
  23. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Opt. Lett. 36, 82 (2011).
    [CrossRef]

2012 (6)

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

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 85023822 (2012).
[CrossRef]

C. Li, C. Huang, H. Liu, and L. Dong, Opt. Lett. 37, 4543 (2012).
[CrossRef]

2011 (5)

2010 (3)

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. A 81, 063807 (2010).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

2009 (3)

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

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, 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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. EI-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)

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

2005 (1)

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]

Bender, C. M.

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

Bendix, O.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Boettcher, S.

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

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, 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, Nature 488, 167 (2012).
[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. A 81, 063807 (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]

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. EI-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Dmitriev, S. V.

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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

EI-Ganainy, R.

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

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[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]

Fleischmann, R.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Ge, L.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85023822 (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]

He, Y.

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

He, Y.-J.

Hu, S.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Hu, W.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Huang, C.

Kartashov, Y. V.

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Opt. Lett. 36, 82 (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.

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

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

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

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

Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, Opt. Lett. 36, 82 (2011).
[CrossRef]

Kottos, T.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

Lakoba, T. I.

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

Li, C.

Li, H.

Liu, H.

Liu, J.

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

Lu, D.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Ma, X.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[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. EI-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Malomed, B. A.

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, 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, Nature 488, 167 (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]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (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. EI-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (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 85023822 (2012).
[CrossRef]

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[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]

Shapiro, B.

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[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]

Sukhorukov, A. 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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Vysloukh, V. A.

Wang, H.

Wang, J.

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

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

Yang, J.

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85023822 (2012).
[CrossRef]

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

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

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

Zheng, L.-X.

Zheng, Y.

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

Zhu, X.

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

X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, Opt. Lett. 36, 2680 (2011).
[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]

Nature (1)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. A (6)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

S. Hu, X. Ma, D. Lu, Y. Zheng, and W. Hu, Phys. Rev. A 85, 043826 (2012).
[CrossRef]

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]

S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85023822 (2012).
[CrossRef]

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

Phys. Rev. Lett. (6)

O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103, 030402 (2009).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, 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, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

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

Z. H. Musslimani, K. G. Makris, R. EI-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]

Stud. Appl. Math. (1)

J. Yang and T. I. Lakoba, Stud. Appl. Math. 118, 153 (2007).
[CrossRef]

Other (1)

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, 2010).

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Figures (5)

Fig. 1.
Fig. 1.

When V0=8 and W0=0.1. (a) The band structure. (Γ, X, and M are the three high symmetry points of the irreducible zone in the first Brillouin zone). (b) The power versus the propagation constant for three-, four-, and six-peak solitons (red shaded regions are the Bloch bands, the solid lines represent the stable regions while the dashed lines represent the unstable regions). (c) and (e) The real (d) and (f) the imaginary parts of the four-peak solitons for μ=6.0 and μ=7.6, respectively.

Fig. 2.
Fig. 2.

For V0=8 and W0=0.1. (a) and (d) are the linear-stability spectra of the four-peak solitons for μ=6.0 and μ=7.6, respectively. (b) and (c) show the profiles (|u|) of the perturbed soliton for μ=6.0 at z=0 and z=500, respectively. (e) and (f) are the profiles (|u|) of the perturbed soliton for μ=7.6 at z=0 and z=480, respectively.

Fig. 3.
Fig. 3.

When V0=8, W0=0.1, and μ=6.0. (a) The real and (b) the imaginary part of the six-peak soliton. (c) and (d) are the real and imaginary parts of the three-peak soliton, respectively.

Fig. 4.
Fig. 4.

For V0=8, W0=0.1, and μ=6.0. (a) and (d) are the linear-stability spectra of the six- and three-peak solitons, respectively. The profiles (|u|) of the perturbed six- and three-peak solitons at (b), (e) z=0 and (c), (f) z=500, respectively.

Fig. 5.
Fig. 5.

When V0=8, W0=0.1, and μ=6.0. The energy flow vector of the four-peak soliton. The contour plotted as the background is the amplitude distribution.

Equations (3)

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iUz+Uxx+UyyV(x,y)U|U|2U=0,
uxx+uyyV0{[cos2(x)+cos2(y)]+iW0[sin(2x)+sin(2y)]}u|u|2uμu=0.
{δg=i[gxx+gyyμgVg2|u|2gu2t],δt=i[txxtyy+μt+V*t+2|u|2t+(u2)*g].

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