Abstract

We identify periodic solitons in nonlocal nonlinear media: multi-hump soliton solutions propagating in a fully periodic fashion. We also demonstrate recurrences and breathers whose evolution is statistically periodic and discuss why some systems support periodic solitons while others do not.

© 2007 Optical Society of America

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References

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  1. A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
    [CrossRef]
  2. E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, Opt. Lett. 29, 283 (2004).
    [CrossRef] [PubMed]
  3. C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  4. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
    [CrossRef]
  5. H. L. Pecseli and J. J. Rasmussen, Plasma Phys. 22, 421 (1980).
    [CrossRef]
  6. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
    [CrossRef] [PubMed]
  7. X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
    [CrossRef]
  8. C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
    [CrossRef] [PubMed]
  9. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]
  10. B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
    [CrossRef]
  11. A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
    [CrossRef]
  12. M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
    [CrossRef]
  13. H. Meng, G. Salamo, M. Shih, and M. Segev, Opt. Lett. 22, 448 (1997).
    [CrossRef] [PubMed]
  14. Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Lett. 30, 3171 (2005).
    [CrossRef] [PubMed]
  15. W. Krolikowski and S. A. Holstrom, Opt. Lett. 22, 369 (1996).
    [CrossRef]

2007 (2)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

2006 (1)

2005 (2)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Lett. 30, 3171 (2005).
[CrossRef] [PubMed]

2004 (3)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, Opt. Lett. 29, 283 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

1999 (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

1997 (3)

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
[CrossRef]

H. Meng, G. Salamo, M. Shih, and M. Segev, Opt. Lett. 22, 448 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (1)

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

1980 (1)

H. L. Pecseli and J. J. Rasmussen, Plasma Phys. 22, 421 (1980).
[CrossRef]

Adamski, A.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Alfassi, B.

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Cambournac, C.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Christodoulides, D. N.

Cohen, O.

C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, Opt. Lett. 31, 3312 (2006).
[CrossRef] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Haelterman, M.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Hewlett, S. J.

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

Holstrom, S. A.

Hutsebaut, X.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Jakubowski, M. H.

M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

W. Krolikowski and S. A. Holstrom, Opt. Lett. 22, 369 (1996).
[CrossRef]

Lange, C. H.

Lederer, F.

Manela, O.

B. Alfassi, C. Rotschild, O. Manela, M. Segev, and D. N. Christodoulides, Opt. Lett. 32, 154 (2007).
[CrossRef]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Meng, H.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

Neyts, K.

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

Pecseli, H. L.

H. L. Pecseli and J. J. Rasmussen, Plasma Phys. 22, 421 (1980).
[CrossRef]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Rasmussen, J. J.

H. L. Pecseli and J. J. Rasmussen, Plasma Phys. 22, 421 (1980).
[CrossRef]

Rotschild, C.

Salamo, G.

Segev, M.

Shih, M.

Snyder, A. W.

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

Squier, R. K.

M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
[CrossRef]

Stegeman, G. I.

Steiglitz, K.

M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Torner, L.

Ultanir, E. A.

Xu, Z.

Opt. Commun. (1)

X. Hutsebaut, C. Cambournac, M. Haelterman, A. Adamski, and K. Neyts, Opt. Commun. 233, 211 (2004).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. E (2)

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[CrossRef]

M. H. Jakubowski, K. Steiglitz, and R. K. Squier, Phys. Rev. E 56, 7267 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Plasma Phys. (1)

H. L. Pecseli and J. J. Rasmussen, Plasma Phys. 22, 421 (1980).
[CrossRef]

Rev. Mod. Phys. (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).
[CrossRef]

Science (1)

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Comparison between the induced potentials in a local Kerr medium (right), a limited-range nonlocal nonlinearity (middle), and an infinite-range nonlocal (thermal) nonlinearity (left). Shown is the potential induced by a soliton of a given width in three samples of different sizes (rectangles). For each sample, the potential well is drawn with the lowest energy level (dotted lines), and the soliton (black curves) self-trapped within it. In local media, the induced potential is not affected by the sample size (as long as the sample is much wider than the soliton). The same occurs for finite range nonlocal nonlinearity, as long as the sample is much wider that the nonlocality range. However, in the nonlocal thermal medium, the potential depth is proportional to the sample width, thus becoming infinitely deep as the sample width goes to infinity.

Fig. 2
Fig. 2

Periodic scalar solitons of the ( 1 + 1 ) D thermal system. a, b, First five periods, for an input relative phase of 0 and π, respectively. c, d, Same periodic solitons propagating over a very large distance. e, f, The induced potential Δ n ( x , z ) for the solitons of a and b. The propagation dynamics is stable and fully periodic at all distances.

Fig. 3
Fig. 3

Two parallel in-phase solitons launched, a, in a saturable nonlinearity with a local response, and, b, in the limited-range nonlocal nonlinearity described by adding the term ϵ Δ n to Eq. (2). b, In both cases, the solitons attract each other and merge (fuse) into a lower mode while emitting radiation.

Fig. 4
Fig. 4

High-order scalar soliton displaying stable quasi-periodic propagation dynamics. The soliton is composed of the first and third modes propagating in a quasi-periodic manner. a, Intensity structure of the scalar high-order soliton. b, The autocorrelation function, calculated for a longer propagation distance, exhibits a statistically periodic structure.

Equations (3)

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i ψ z + 1 2 k 2 ψ + k n 0 Δ n ψ = 0
2 Δ n = α ψ 2 .
Δ n ( x , z ) = α d d x x ψ ( x , z ) 2 d x x α d P x C M + α d P

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