Abstract

We address the existence and stability of two-dimensional solitons in optical or matter-wave media, which are supported by purely nonlinear lattices in the form of a periodic array of cylinders with self-focusing nonlinearity, embedded into a linear material. We show that such lattices can stabilize two-dimensional solitons against collapse. We also found that stable multipoles and vortex solitons are also supported by nonlinear lattices, provided that the nonlinearity exhibits saturation.

© 2009 Optical Society of America

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  1. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
    [CrossRef]
  2. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
    [CrossRef]
  3. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
    [CrossRef] [PubMed]
  4. R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
    [CrossRef] [PubMed]
  5. B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).
  6. J. Yang and Z. H. Musslimani, Opt. Lett. 28, 2094 (2003).
    [CrossRef] [PubMed]
  7. H. Sakaguchi and B. A. Malomed, Electron. Lett. 72, 698 (2005).
  8. Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
    [CrossRef]
  9. T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
    [CrossRef]
  10. H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002).
    [CrossRef]
  11. H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005).
    [CrossRef]
  12. G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
    [CrossRef]
  13. J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
    [CrossRef] [PubMed]
  14. F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
    [CrossRef]
  15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1747 (2008).
    [CrossRef] [PubMed]
  16. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 2173 (2008).
    [CrossRef] [PubMed]
  17. Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
    [CrossRef] [PubMed]
  18. Y. Kominis and K. Hizanidis, Opt. Lett. 16, 12124 (2008).
  19. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
    [CrossRef] [PubMed]
  20. B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
    [CrossRef]
  21. H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006).
    [CrossRef]

2008 (5)

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
[CrossRef]

Y. Kominis and K. Hizanidis, Opt. Lett. 16, 12124 (2008).

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1747 (2008).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 2173 (2008).
[CrossRef] [PubMed]

2007 (2)

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
[CrossRef]

2006 (5)

B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006).
[CrossRef]

G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
[CrossRef]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

2005 (3)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Electron. Lett. 72, 698 (2005).

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005).
[CrossRef]

2004 (2)

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

2003 (3)

J. Yang and Z. H. Musslimani, Opt. Lett. 28, 2094 (2003).
[CrossRef] [PubMed]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

2002 (1)

H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002).
[CrossRef]

Abdullaev, F.

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
[CrossRef]

Abdumalikov, A.

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
[CrossRef]

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).

Belmonte-Beitia, J.

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

Chen, Z.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Christodoulides, D. N.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

Denz, C.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Dong, G.

T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
[CrossRef]

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Fibich, G.

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
[CrossRef]

Fischer, R.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

Galimzyanov, R.

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
[CrossRef]

Hizanidis, K.

Y. Kominis and K. Hizanidis, Opt. Lett. 16, 12124 (2008).

Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

Ilan, B.

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

Kartashov, Y. V.

Kivshar, Y. S.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Kominis, Y.

Y. Kominis and K. Hizanidis, Opt. Lett. 16, 12124 (2008).

Y. Kominis and K. Hizanidis, Opt. Lett. 31, 2888 (2006).
[CrossRef] [PubMed]

Krolikowski, W.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Malomed, B. A.

T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Electron. Lett. 72, 698 (2005).

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005).
[CrossRef]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).

Martin, H.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

Mayteevarunyoo, T.

T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
[CrossRef]

Mihalache, D.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Musslimani, Z. H.

Neshev, D. N.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Perez-Garcia, V. M.

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

Saito, H.

H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Electron. Lett. 72, 698 (2005).

Salerno, M.

B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).

Segev, M.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

Sivan, Y.

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
[CrossRef]

Sukhorukov, A. A.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 1747 (2008).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Lett. 33, 2173 (2008).
[CrossRef] [PubMed]

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

Torres, P. J.

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

Träger, D.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Ueda, M.

H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002).
[CrossRef]

Vekslerchik, V.

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

Vysloukh, V. A.

Weinstein, M. I.

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
[CrossRef]

Wise, F.

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Yang, J.

Electron. Lett. (2)

H. Sakaguchi and B. A. Malomed, Electron. Lett. 72, 698 (2005).

B. B. Baizakov, B. A. Malomed, and M. Salerno, Electron. Lett. 63, 642 (2003).

J. Opt. B (1)

B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).
[CrossRef]

Nature (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Nature 422, 148 (2003).
[CrossRef]

Opt. Lett. (5)

Phys. Lett. A (1)

F. Abdullaev, A. Abdumalikov, and R. Galimzyanov, Phys. Lett. A 367, 149 (2007).
[CrossRef]

Phys. Rev. A (2)

T. Mayteevarunyoo, B. A. Malomed, and G. Dong, Phys. Rev. A 78, 053601 (2008).
[CrossRef]

H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002).
[CrossRef]

Phys. Rev. E (4)

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 72, 046610 (2005).
[CrossRef]

Y. Sivan, G. Fibich, B. Ilan, and M. I. Weinstein, Phys. Rev. E 78, 046602 (2008).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, Phys. Rev. E 74, 066615 (2006).
[CrossRef]

H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006).
[CrossRef]

Phys. Rev. Lett. (4)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, Phys. Rev. Lett. 92, 123902 (2004).
[CrossRef] [PubMed]

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

J. Belmonte-Beitia, V. M. Perez-Garcia, V. Vekslerchik, and P. J. Torres, Phys. Rev. Lett. 98, 064102 (2007).
[CrossRef] [PubMed]

Physica D (1)

G. Fibich, Y. Sivan, and M. I. Weinstein, Physica D 217, 31 (2006).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Profiles of the absolute value of the field for (a) and (b) fundamental solitons, (c) and (d) dipoles, and (e) and (f) off-site vortices. (a), (c), and (e) correspond to b = 0.05 while panels (b), (d), and (f) correspond to b = 7 . Here and in Figs. 2d, 3, w s = 3 and S = 0.1 .

Fig. 2
Fig. 2

(a) Soliton power versus b for several values of w s in the cubic medium. (b) Minimum power versus lattice spacing w s . The horizontal dashed lines in (a) and (b) correspond to the Townes soliton power, U T 5.85 . (c) Real part of the perturbation growth rate versus b at w s = 3 . (d) Power versus b in the saturable medium. Circles in (d) correspond to solitons from Figs. 1a, 1b.

Fig. 3
Fig. 3

(a) Power versus b for dipoles and vortices. The real part of the perturbation growth rate versus b for (b) dipoles and (c) vortices. Circles in (a) correspond to solitons from Figs. 1c, 1d, 1e, 1f.

Equations (3)

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i q ξ = 1 2 ( 2 q η 2 + 2 q ζ 2 ) + σ ( η , ζ ) q q 2 1 + S q 2 ,
δ u = 1 2 ( 2 v η 2 + 2 v ζ 2 ) + b v + σ ( η , ζ ) v w 2 + S w 4 ( 1 + S w 2 ) 2 ,
δ v = + 1 2 ( 2 u η 2 + 2 u ζ 2 ) b u σ ( η , ζ ) u 3 w 2 + S w 4 ( 1 + S w 2 ) 2 .

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