Abstract

We introduce two different systems of coupled-mode equations to describe the interaction of two waves coupled by the Bragg reflection in the presence of saturable nonlinearity. The basic model assumes the ordinary linear coupling between the modes. It may be realized as a photorefractive waveguide, with a Bragg lattice permanently written in its cladding. We demonstrate the presence of a cutoff point in the system’s bandgap, with gap solitons existing only on one side of it. Close to this point, the soliton’s norm diverges with power 32. The soliton family between the cutoff point and the edge of the bandgap is stable. In this model, stationary bound states of two in-phase solitons are found too, but they are unstable, transforming themselves into breathers. Another model assumes a photoinduced longitudinal bulk grating, with the corresponding intermode coupling subject to saturation along with the nonlinearity. In that model, another cutoff point is found, with the soliton’s norm diverging near it with power 2. Solitons are stable in this model too (while it does not give rise to two-soliton bound states). Collisions between moving solitons are always quasi-elastic, in either model.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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  8. A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
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  16. P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, "Discrete gap solitons in a diffraction-managed waveguide array," Eur. Phys. J. D 23, 421-236 (2003).
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  17. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef] [PubMed]
  18. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 093904 (2004).
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  20. F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005).
    [CrossRef] [PubMed]
  21. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
    [CrossRef] [PubMed]
  22. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, "Controlled generation and steering of spatial gap solitons," Phys. Rev. Lett. 93, 083905 (2004).
    [CrossRef] [PubMed]
  23. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
    [CrossRef] [PubMed]
  24. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
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  25. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, "Two-dimensional optical lattice solitons," Phys. Rev. Lett. 91, 213906 (2003).
    [CrossRef] [PubMed]
  26. G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
    [CrossRef] [PubMed]
  27. X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083904 (2006).
    [CrossRef] [PubMed]
  28. B. B. Baizakov, B. A. Malomed, and M. Salerno, "Matter-wave solitons in radially periodic potentials," Phys. Rev. A 70, 053613 (2004).
    [CrossRef]
  29. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904 (2004).
    [CrossRef] [PubMed]
  30. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile vortex solitons in Bessel optical lattices," Phys. Rev. Lett. 94, 043902 (2005).
    [CrossRef] [PubMed]
  31. B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
    [CrossRef]
  32. O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
    [CrossRef]
  33. M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, andC. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025601 (2003).
    [CrossRef]
  34. K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
    [CrossRef]
  35. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
    [CrossRef] [PubMed]
  36. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
    [CrossRef] [PubMed]
  37. S. Lan, M. Shih, and M. Segev, "Self-trapping of one-dimensional and two-dimensional optical beams and induced waveguides in photorefractive KNbO3," Opt. Lett. 22, 1467-1469 (1997).
    [CrossRef]
  38. J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 155, 247-252 (2001).
    [CrossRef]
  39. W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Formation of a standing-light pulse through collision of gap solitons," Phys. Rev. E 68, 026609 (2003).
    [CrossRef]
  40. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, "Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression," J. Lightwave Technol. 15, 1303-1313 (1997).
    [CrossRef]
  41. M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
    [CrossRef]
  42. T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, "Solitary waves in Bragg gratings with a quadratic nonlinearity," Phys. Rev. E 55, 4730-4739 (1997).
    [CrossRef]
  43. Y. Leitner and B. A. Malomed, "Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity," Phys. Rev. E 71, 057601 (2005).
    [CrossRef]
  44. E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, "Stability of multihump optical solitons," Phys. Rev. Lett. 83, 296-299 (1999).
    [CrossRef]

2006 (1)

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

2005 (5)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile vortex solitons in Bessel optical lattices," Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef] [PubMed]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

Y. Leitner and B. A. Malomed, "Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity," Phys. Rev. E 71, 057601 (2005).
[CrossRef]

G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. K. Efremidis, "Spatial photonics in nonlinear waveguide arrays," Opt. Express 13, 1780-1796 (2005).
[CrossRef] [PubMed]

F. Chen, M. Stepic, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, "Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays," Opt. Express 13, 4314-4324 (2005).
[CrossRef] [PubMed]

2004 (5)

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Matter-wave solitons in radially periodic potentials," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, "Controlled generation and steering of spatial gap solitons," Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

2003 (8)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, "Discrete gap solitons in a diffraction-managed waveguide array," Eur. Phys. J. D 23, 421-236 (2003).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, andC. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, "Two-dimensional optical lattice solitons," Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Formation of a standing-light pulse through collision of gap solitons," Phys. Rev. E 68, 026609 (2003).
[CrossRef]

2002 (3)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
[CrossRef]

A. A. Sukhorukov and Yu. S. Kivshar, "Discrete gap solitons in modulated waveguide arrays," Opt. Lett. 27, 2112-2114 (2002).
[CrossRef]

2001 (1)

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 155, 247-252 (2001).
[CrossRef]

1999 (2)

B. J. Eggleton, C. M. De Sterke, and R. E. Slusher, "Bragg solitons in the nonlinear Schrödinger limit: experiment and theory," J. Opt. Soc. Am. B 16, 587-599 (1999).
[CrossRef]

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, "Stability of multihump optical solitons," Phys. Rev. Lett. 83, 296-299 (1999).
[CrossRef]

1998 (3)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Three-wave gap solitons in waveguides with quadratic nonlinearity," Phys. Rev. E 58, 6708-6722 (1998).
[CrossRef]

1997 (3)

S. Lan, M. Shih, and M. Segev, "Self-trapping of one-dimensional and two-dimensional optical beams and induced waveguides in photorefractive KNbO3," Opt. Lett. 22, 1467-1469 (1997).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, "Solitary waves in Bragg gratings with a quadratic nonlinearity," Phys. Rev. E 55, 4730-4739 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, "Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression," J. Lightwave Technol. 15, 1303-1313 (1997).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1995 (1)

Yu. S. Kivshar, "Gap solitons due to cascading," Phys. Rev. E 51, 1613--1615 (1995).
[CrossRef]

1994 (1)

B. A. Malomed and R. S. Tasgal, "Vibration modes of a gap soliton in a nonlinear optical medium," Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

1993 (3)

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

J. Feng, "Alternative scheme for studying gap solitons in an infinite periodic Kerr medium," Opt. Lett. 18, 1302-1304 (1993).
[CrossRef] [PubMed]

R. F. Nabiev, P. Yeh, and D. Botez, "Spatial gap solitons in periodic nonlinear structures," Opt. Lett. 18, 1612-1614 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

1989 (2)

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structure," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

1973 (1)

M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
[CrossRef]

M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

Aitchison, J. S.

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Atai, J.

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 155, 247-252 (2001).
[CrossRef]

Baizakov, B. B.

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Matter-wave solitons in radially periodic potentials," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

Bartal, G.

Belic, M.

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, andC. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025601 (2003).
[CrossRef]

Botez, D.

Buljan, H.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
[CrossRef]

Chen, F.

Chen, Z.

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Christodoulides, D. N.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, "Observation of discrete gap solitons in binary waveguide arrays," Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, "Two-dimensional optical lattice solitons," Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structure," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Formation of a standing-light pulse through collision of gap solitons," Phys. Rev. E 68, 026609 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Three-wave gap solitons in waveguides with quadratic nonlinearity," Phys. Rev. E 58, 6708-6722 (1998).
[CrossRef]

Cohen, O.

Conti, C.

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

Crosignani, B.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

De Rossi, A.

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

De Sterke, C. M.

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K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
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E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, "Stability of multihump optical solitons," Phys. Rev. Lett. 83, 296-299 (1999).
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N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, "Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression," J. Lightwave Technol. 15, 1303-1313 (1997).
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G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
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Sharp, E. J.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
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G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
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D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 053902 (2003).
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B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
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E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, "Stability of multihump optical solitons," Phys. Rev. Lett. 83, 296-299 (1999).
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B. J. Eggleton, C. M. De Sterke, and R. E. Slusher, "Bragg solitons in the nonlinear Schrödinger limit: experiment and theory," J. Opt. Soc. Am. B 16, 587-599 (1999).
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Stepic, M.

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B. A. Malomed and R. S. Tasgal, "Vibration modes of a gap soliton in a nonlinear optical medium," Phys. Rev. E 49, 5787-5796 (1994).
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Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile vortex solitons in Bessel optical lattices," Phys. Rev. Lett. 94, 043902 (2005).
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M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
[CrossRef]

M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
[CrossRef]

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile vortex solitons in Bessel optical lattices," Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

Wabnitz, S.

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Wang, X.

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Yariv, A.

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Yeh, P.

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

Eur. Phys. J. D (1)

P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, "Discrete gap solitons in a diffraction-managed waveguide array," Eur. Phys. J. D 23, 421-236 (2003).
[CrossRef]

J. Lightwave Technol. (1)

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, "Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression," J. Lightwave Technol. 15, 1303-1313 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (6)

Phys. Lett. A (2)

A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

J. Atai and B. A. Malomed, "Families of Bragg-grating solitons in a cubic-quintic medium," Phys. Lett. A 155, 247-252 (2001).
[CrossRef]

Phys. Rev. A (1)

B. B. Baizakov, B. A. Malomed, and M. Salerno, "Matter-wave solitons in radially periodic potentials," Phys. Rev. A 70, 053613 (2004).
[CrossRef]

Phys. Rev. E (10)

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, "Discrete solitons in photorefractive optically induced photonic lattices," Phys. Rev. E 66, 046602 (2002).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Formation of a standing-light pulse through collision of gap solitons," Phys. Rev. E 68, 026609 (2003).
[CrossRef]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, andC. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

B. A. Malomed and R. S. Tasgal, "Vibration modes of a gap soliton in a nonlinear optical medium," Phys. Rev. E 49, 5787-5796 (1994).
[CrossRef]

Yu. S. Kivshar, "Gap solitons due to cascading," Phys. Rev. E 51, 1613--1615 (1995).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, "Three-wave gap solitons in waveguides with quadratic nonlinearity," Phys. Rev. E 58, 6708-6722 (1998).
[CrossRef]

T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, "Solitary waves in Bragg gratings with a quadratic nonlinearity," Phys. Rev. E 55, 4730-4739 (1997).
[CrossRef]

Y. Leitner and B. A. Malomed, "Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity," Phys. Rev. E 71, 057601 (2005).
[CrossRef]

B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005).
[CrossRef]

Phys. Rev. Lett. (15)

E. A. Ostrovskaya, Y. S. Kivshar, D. V. Skryabin, and W. J. Firth, "Stability of multihump optical solitons," Phys. Rev. Lett. 83, 296-299 (1999).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons," Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, "Gap solitons in waveguide arrays," Phys. Rev. Lett. 92, 093904 (2004).
[CrossRef] [PubMed]

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, "Vibrations and oscillatory instabilities of gap solitons," Phys. Rev. Lett. 80, 5117-5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, "Stability, multistability, and wobbling of optical gap solitons," Phys. Rev. Lett. 81, 85-88 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, "Slow Bragg solitons in nonlinear periodic structure," Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, "Two-dimensional optical lattice solitons," Phys. Rev. Lett. 91, 213906 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of discrete solitons in optically induced real time waveguide arrays," Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Yu. S. Kivshar, "Controlled generation and steering of spatial gap solitons," Phys. Rev. Lett. 93, 083905 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Stable ring-profile vortex solitons in Bessel optical lattices," Phys. Rev. Lett. 94, 043902 (2005).
[CrossRef] [PubMed]

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices," Phys. Rev. Lett. 96, 083904 (2006).
[CrossRef] [PubMed]

Sov. J. Radiophys. Quantum Electr. (1)

M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Sov. J. Radiophys. Quantum Electr. 16, 783-789 (1973) M. G. Vakhitov and A. A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation,"[Izv. Vuz. Radiofiz. 16, 1020-1028 (1973)].
[CrossRef]

Other (3)

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic, 2003).

C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E.Wolf, ed. (North-Holland, 1994), Vol. XXXIII, Chap. III, pp. 203-260.

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Figures (12)

Fig. 1
Fig. 1

Typical examples of stable straight (zero-velocity) solitons found in the Bragg-grating model with the nonlinearity saturation. The solitons are obtained as numerical solutions of Eq. (13) with κ = 1 . (a) and (b) correspond to ω = 1.8 and 1.1 , i.e., to the solitons located, respectively, close to the upper edge and center of the bandgap. Here and in Figs. 3, 5, profiles marked by “cubic” display, for the sake of comparison, exact solutions for the gap solitons in the standard model, which is based on Eqs. (1) with the cubic nonlinearity [to make them as close as possible to Eqs. (5), we set σ = 1 in Eqs. (1)]. The latter solutions were taken also for κ = 1 , with the substitution of ω ω 1 , see Eq. (7).

Fig. 2
Fig. 2

Norm of the soliton, defined as per Eq. (10), versus ω for the family of numerically found straight (zero-velocity) soliton solutions to Eq. (13), with κ = 1.5 . The family exists in interval 1.5 < ω < 2.5 , which corresponds to Eq. (14). For ω 1.5 0 , the norm diverges, as the amplitude of the real part of the solution and its width tend to become infinitely large; see Fig. 1b. For the sake of comparison, a curve showing the norm of the family of exact solitons in the standard model, based on Eqs. (1) with σ = 1 , multiplied by 10 (otherwise, it would be almost invisible) is included too.

Fig. 3
Fig. 3

Examples of stable tilted (moving) solitons in the BG model with the saturable nonlinearity, for κ = 1 , c = 0.1 , and ω = 1.8 (a) and ω = 1.1 (b), i.e., respectively, close to the upper edge and center of the bandgap, cf. Fig. 1. For comparison, exact solutions for the moving solitons found, at the same parameters, in the standard model based on Eqs. (1), with σ = 1 , are included too.

Fig. 4
Fig. 4

Momentum of the moving (tilted) solitons, found as per Eq. (11), plotted versus their velocity, for κ = 1 . The intrinsic frequency of the solitons is fixed at (a) ω = 1.8 and (b) ω = 1.1 . The P ( c ) dependences for exact solitons in the standard model, based on Eqs. (1), with σ = 1 , are included for the comparison.

Fig. 5
Fig. 5

Examples of stable straight (zero-velocity) solitons found in the BG model with the coupling saturation, based on Eqs. (6) with κ = 1 : (a) ω = 0.18 ; (b) ω = 1.8 . For comparison, solitons in the standard BG model, based on Eqs. (1) and found at the corresponding values of parameters (in particular, σ = 1 ), are shown too.

Fig. 6
Fig. 6

Norm of the soliton defined as per Eq. (10), is plotted versus ω for the family of numerically found straight (zero-velocity-) soliton solutions of Eq. (18), with κ = 1 . The family exists in the interval [Eq. (19)], which, in the present case, is 0 < ω < 2 , coinciding with the entire bandgap [Eq. (8)] (for κ > 1 , the existence interval is smaller than the bandgap; see text). For ω 0 , the norm, amplitude and width of the soliton diverge in accordance with analytical predictions [Eq. (20)]. For comparison, a curve showing the norm of the family of exact solitons in the standard model, based on Eqs. (1) with σ = 1 and κ = 1 , multiplied by 10 (to make it visible), is included too, cf. Fig. 2.

Fig. 7
Fig. 7

Examples of stable moving (tilted) solitons in the model with the saturation of the nonlinearity, for κ = 1 : (a) c = 0.5 , ω = 1.8 and (b) c = 0.2 , ω = 0.18 , i.e., near the upper and lower edge of the bandgap, respectively; cf. Fig. 3. For comparison, included are also exact solutions for the moving solitons (at the corresponding values of the parameters) in the standard model based on Eqs. (1), with σ = 1 .

Fig. 8
Fig. 8

Evolution of a perturbed straight (zero-velocity) soliton in the model based on Eqs. (5), for κ = 1 and ω = 1.8 , i.e., when the unperturbed soliton is the same as in Fig. 1. In this figure and in other figures that present results of direct simulations (see below) only the U component is displayed, as the evolution of its V counterpart seems quite similar, in all cases.

Fig. 9
Fig. 9

Evolution of the tilted (moving) solitons from Figs. 3a, 3b, to which a 2 % amplitude perturbation was added.

Fig. 10
Fig. 10

Typical example of a DH stationary pattern found from Eq. (13) at ω = 3.4 and κ = 2.5 . For comparison, a fundamental (SH) soliton, found at the same values of parameters, is shown too.

Fig. 11
Fig. 11

Transformation of the unstable bound state of two in-phase solitons into a breather, at κ = 2.5 and ω = 3.4 (the initial bound state is the same as in Fig. 10).

Fig. 12
Fig. 12

Examples of collisions between identical solitons moving at velocities ± c : (a) the inelastic collision at c = 0.1 , produced by simulations of the standard BG model, based on Eqs. (1) with σ = 0.5 ; (b) quasi-elastic collision at the same velocities, c = 0.1 , revealed by simulations of Eqs. (5). In both cases, the equations were taken with κ = 1 , and the colliding solitons had intrinsic frequencies (a) ω = 0.6675 and (b) ω = 1.8 .

Equations (30)

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i U t + i U x + ( σ U 2 + V 2 ) U + κ V = 0 ,
i V t i V x + ( σ V 2 + U 2 ) V + κ U = 0 ,
i E z + 1 2 2 E x 2 E 1 + I 0 cos 2 ( K x ) + E 2 = 0 ,
i U z + i K U x = ( U V ) 1 + I 0 ( 1 + U V 2 ) + 2 ( U 2 + V 2 ) ,
i V z i K V x = ( V U ) 1 + I 0 ( 1 + U V 2 ) + 2 ( U 2 + V 2 ) .
i U z + i K U x = ( U V ) I 0 ( 1 + U 2 + V 2 ) ,
i V z i K V x = ( V U ) I 0 ( 1 + U 2 + V 2 )
i U t + i U x U 1 + U 2 + V 2 + κ V = 0 ,
i V t i V x V 1 + U 2 + V 2 + κ U = 0 .
i U t + i U x U κ V 1 + U 2 + V 2 = 0 ,
i V t i V x V κ U 1 + U 2 + V 2 = 0 .
ω = 1 ± κ 2 + q 2
1 κ < ω < 1 + κ
i k [ ( cos θ ) U z + ( sin θ ) U x ] , i k [ ( cos θ ) V z ( sin θ ) V x ]
E = + [ U ( x ) 2 + V ( x ) 2 ] d x .
H = 1 2 + [ i ( U U x * V V x * ) + ln ( 1 + U 2 + V 2 ) 2 κ U * V ] d x + c.c. ,
P = i + ( U U x * + V V x * ) d x .
{ U , V } = { u ( x ) , v ( x ) } exp ( i ω t ) .
ω u + i d u d x u 1 + 2 u 2 κ u * = 0 .
κ < ω < 1 + κ ,
A W ( ω κ ) 1 2 ,
E A 2 W ( ω κ ) 3 2 ,
ω u + ( 1 c ) i d u d ξ u 1 + u 2 + v 2 + κ v = 0 ,
ω v ( 1 + c ) i d v d ξ v 1 + v 2 + u 2 + κ u = 0 ,
ω u + i d u d x u + κ u * 1 + 2 u 2 = 0 .
0 < ω < 1 + κ .
A ω 1 2 , W ω 1 , E ω 2 .
{ U ( x , t ) V ( x , t ) } = [ { u 0 ( x ) v 0 ( x ) } + { u 1 ( x ) v 1 ( x ) } e i λ t ] e i ω t ,
( ω + i d d x ) u 1 u 1 1 + u 0 2 + v 0 2 + ( u 1 u 0 * + u 0 u 1 * ) + v 1 v 0 * + v 0 v 1 * ( 1 + u 0 2 + v 0 2 ) 2 u 0 + κ v 1 = λ u 1 ,
( ω i d d x ) v 1 v 1 1 + v 0 2 + u 0 2 + ( v 1 v 0 * + v 0 v 1 * ) + u 1 u 0 * + u 0 u 1 * ( 1 + v 0 2 + u 0 2 ) 2 v 0 + κ u 1 = λ v 1 .

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