Abstract

We investigate numerically and experimentally the influence of coupling disorder on the self-trapping dynamics in nonlinear one-dimensional optical waveguide arrays. The existence of a lower and upper bound of the effective average propagation constant allows for a generalized definition of the threshold power for the onset of soliton localization. When compared to perfectly ordered systems, this threshold is found to decrease in the presence of coupling disorder.

© 2013 Optical Society of America

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  1. S. Flach and A. Gorbach, Phys. Rep. 467(1–3), 1 (2008).
    [CrossRef]
  2. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
    [CrossRef]
  3. V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
    [CrossRef]
  4. M. I. Molina and G. P. Tsironis, Physica D 65, 267 (1993).
    [CrossRef]
  5. M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
    [CrossRef]
  6. C. A. Bustamante and M. I. Molina, Phys. Rev. B 62, 15287 (2000).
    [CrossRef]
  7. J. Meier, J. Hudock, D. N. Christodoulides, G. I. Stegeman, H. Y. Yang, G. Salamo, R. Morandotti, J. S. Aitchison, and Y. Silberberg, J. Opt. Soc. Am. B 22, 1432 (2005).
    [CrossRef]
  8. A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, and A. Tnnermann, Opt. Express 13, 10552 (2005).
    [CrossRef]
  9. A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, Opt. Express 13, 1808 (2005).
    [CrossRef]
  10. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
    [CrossRef]
  11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
    [CrossRef]
  12. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
    [CrossRef]
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    [CrossRef]
  14. A. Szameit, Y. V. Kartashov, P. Zeil, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and A. Tünnermann, Opt. Lett. 35, 1172 (2010).
    [CrossRef]
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    [CrossRef]
  16. A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008).
    [CrossRef]
  17. M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
    [CrossRef]
  18. J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
    [CrossRef]
  19. U. Naether, S. Rojas-Rojas, A. J. Martínez, S. Stützer, A. Tünnermann, S. Nolte, M. I. Molina, R. A. Vicencio, and A. Szameit, Opt. Express 21, 927 (2013).
    [CrossRef]
  20. G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
    [CrossRef]
  21. S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
    [CrossRef]
  22. T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
    [CrossRef]
  23. U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).
  24. M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
    [CrossRef]

2013

2011

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
[CrossRef]

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

2010

2009

S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
[CrossRef]

2008

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

S. Flach and A. Gorbach, Phys. Rep. 467(1–3), 1 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

2007

2005

2004

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

2000

C. A. Bustamante and M. I. Molina, Phys. Rev. B 62, 15287 (2000).
[CrossRef]

1995

M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
[CrossRef]

1993

M. I. Molina and G. P. Tsironis, Physica D 65, 267 (1993).
[CrossRef]

1986

V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
[CrossRef]

1981

C. M. Soukoulis and E. N. Economou, Phys. Rev. B 24, 5698 (1981).
[CrossRef]

1958

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Aitchison, J. S.

Anderson, P. W.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, Opt. Express 13, 1808 (2005).
[CrossRef]

Aubry, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Bahrampour, A. R.

M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
[CrossRef]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Bartelt, H.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Blömer, D.

Bodyfelt, J. D.

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

Brzdakiewicz, K.

Burghoff, J.

Bustamante, C. A.

C. A. Bustamante and M. I. Molina, Phys. Rev. B 62, 15287 (2000).
[CrossRef]

Campbell, D. K.

V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
[CrossRef]

Christodoulides, D. N.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

J. Meier, J. Hudock, D. N. Christodoulides, G. I. Stegeman, H. Y. Yang, G. Salamo, R. Morandotti, J. S. Aitchison, and Y. Silberberg, J. Opt. Soc. Am. B 22, 1432 (2005).
[CrossRef]

Dreisow, F.

Economou, E. N.

C. M. Soukoulis and E. N. Economou, Phys. Rev. B 24, 5698 (1981).
[CrossRef]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Flach, S.

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
[CrossRef]

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
[CrossRef]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

S. Flach and A. Gorbach, Phys. Rep. 467(1–3), 1 (2008).
[CrossRef]

Fratalocchi, A.

Gligoric, G.

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

Golshani, M.

M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
[CrossRef]

Gorbach, A.

S. Flach and A. Gorbach, Phys. Rep. 467(1–3), 1 (2008).
[CrossRef]

Guzmán-Silva, D.

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

Heinrich, M.

Hörnquist, M.

M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
[CrossRef]

Hudock, J.

Ivanchenko, M. V.

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
[CrossRef]

Johansson, M.

M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
[CrossRef]

Karpierz, M.

Kartashov, Y. V.

Keil, R.

Kenkre, V. M.

V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
[CrossRef]

Khazaei Nezhad, M.

M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
[CrossRef]

Kobelke, J.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Komineas, S.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

Kopidakis, G.

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

Krimer, D. O.

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
[CrossRef]

Lahini, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Laptyeva, T. V.

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Mahdavi, S. M.

M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
[CrossRef]

Martínez, A. J.

U. Naether, S. Rojas-Rojas, A. J. Martínez, S. Stützer, A. Tünnermann, S. Nolte, M. I. Molina, R. A. Vicencio, and A. Szameit, Opt. Express 21, 927 (2013).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

Meier, J.

Molina, M. I.

U. Naether, S. Rojas-Rojas, A. J. Martínez, S. Stützer, A. Tünnermann, S. Nolte, M. I. Molina, R. A. Vicencio, and A. Szameit, Opt. Express 21, 927 (2013).
[CrossRef]

C. A. Bustamante and M. I. Molina, Phys. Rev. B 62, 15287 (2000).
[CrossRef]

M. I. Molina and G. P. Tsironis, Physica D 65, 267 (1993).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

J. Meier, J. Hudock, D. N. Christodoulides, G. I. Stegeman, H. Y. Yang, G. Salamo, R. Morandotti, J. S. Aitchison, and Y. Silberberg, J. Opt. Soc. Am. B 22, 1432 (2005).
[CrossRef]

Naether, U.

U. Naether, S. Rojas-Rojas, A. J. Martínez, S. Stützer, A. Tünnermann, S. Nolte, M. I. Molina, R. A. Vicencio, and A. Szameit, Opt. Express 21, 927 (2013).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

Nolte, S.

Pertsch, T.

Peschel, U.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Pikovsky, A. S.

A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008).
[CrossRef]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Riklund, R.

M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
[CrossRef]

Rojas-Rojas, S.

Salamo, G.

Schreiber, T.

Schuster, K.

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Schwartz, T.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Shepelyansky, D. L.

A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008).
[CrossRef]

Silberberg, Y.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

J. Meier, J. Hudock, D. N. Christodoulides, G. I. Stegeman, H. Y. Yang, G. Salamo, R. Morandotti, J. S. Aitchison, and Y. Silberberg, J. Opt. Soc. Am. B 22, 1432 (2005).
[CrossRef]

Skokos, Ch.

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
[CrossRef]

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis and E. N. Economou, Phys. Rev. B 24, 5698 (1981).
[CrossRef]

Stegeman, G. I.

Stützer, S.

Szameit, A.

Tnnermann, A.

Tsironis, G. P.

M. I. Molina and G. P. Tsironis, Physica D 65, 267 (1993).
[CrossRef]

Tuennermann, A.

Tünnermann, A.

Vicencio, R. A.

U. Naether, S. Rojas-Rojas, A. J. Martínez, S. Stützer, A. Tünnermann, S. Nolte, M. I. Molina, R. A. Vicencio, and A. Szameit, Opt. Express 21, 927 (2013).
[CrossRef]

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

Yang, H. Y.

Zeil, P.

Int. J. Bifurc. Chaos

J. D. Bodyfelt, T. V. Laptyeva, G. Gligoric, D. O. Krimer, Ch. Skokos, and S. Flach, Int. J. Bifurc. Chaos 21, 2107 (2011).
[CrossRef]

J. Opt. Soc. Am. B

Nature

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, Nature 446, 52 (2007).
[CrossRef]

Opt. Commun.

M. Khazaei Nezhad, M. Golshani, A. R. Bahrampour, and S. M. Mahdavi, Opt. Commun. 294, 299 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

S. Flach and A. Gorbach, Phys. Rep. 467(1–3), 1 (2008).
[CrossRef]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463(1–3), 1 (2008).
[CrossRef]

Phys. Rev.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. B

V. M. Kenkre and D. K. Campbell, Phys. Rev. B 34, 4959 (1986).
[CrossRef]

M. Johansson, M. Hörnquist, and R. Riklund, Phys. Rev. B 52, 231 (1995).
[CrossRef]

C. A. Bustamante and M. I. Molina, Phys. Rev. B 62, 15287 (2000).
[CrossRef]

C. M. Soukoulis and E. N. Economou, Phys. Rev. B 24, 5698 (1981).
[CrossRef]

Phys. Rev. Lett.

A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett. 100, 094101 (2008).
[CrossRef]

M. V. Ivanchenko, T. V. Laptyeva, and S. Flach, Phys. Rev. Lett. 107, 240602 (2011).
[CrossRef]

G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett. 100, 084103 (2008).
[CrossRef]

S. Flach, D. O. Krimer, and Ch. Skokos, Phys. Rev. Lett. 102, 024101 (2009).
[CrossRef]

T. Pertsch, U. Peschel, J. Kobelke, K. Schuster, H. Bartelt, S. Nolte, A. Tünnermann, and F. Lederer, Phys. Rev. Lett. 93, 053901 (2004).
[CrossRef]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, Phys. Rev. Lett. 100, 013906 (2008).
[CrossRef]

Physica D

M. I. Molina and G. P. Tsironis, Physica D 65, 267 (1993).
[CrossRef]

Other

U. Naether, A. J. Martínez, D. Guzmán-Silva, M. I. Molina, and R. A. Vicencio,Phys. Rev. E, arXiv:1212.2936 (2012).

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Figures (3)

Fig. 1.
Fig. 1.

Power-dependent effective propagation constant. Shown is the mean λe (solid black graph) and λe|z=N4 (solid gray) obtained from central single-site excitations in a homogeneous lattice (Cn=1) with N=221. Inset: R versus P.

Fig. 2.
Fig. 2.

(a) Transition between linear diffraction and nonlinear localization for varying off-diagonal disorder. Shown is λeP/2 for Wc=0 (solid black), Wc=0.1 (dashed blue), and Wc=0.6 (dotted red). (b) Ensemble averages and standard error of Pth obtained for Δ=0.05 for off- (on-)diagonal disorder with full (dashed) lines.

Fig. 3.
Fig. 3.

(a) Experimental mean output intensity distributions at z=1.28π/2C¯ for σ=0C¯ [top], σ=0.4C¯ [center], and σ=0.8C¯ [bottom]. (b) Transition region: power-dependent deviation from linear behavior λeP/¯2 for σ=0C¯ (solid black), σ=0.4C¯ (dashed blue), and σ=0.8C¯ (dotted red). Inset: λe remains confined to the interval P/2λeP.

Equations (6)

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

idundz+εnun+Cnun+1+Cn1un1+γ|un|2un=0,
Hn[Cn(un+1*un+un*un+1)+|un|42],
RP2n|un|4,
λen|un|2=n[Cn(un+1*un+un*un+1)+|un|4].
λeP=H+12n|un|4=H+12P2R.
λe=P2(1+1R).

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