Abstract

We present an overview of the properties of nonlinear guided waves and (bright and dark) spatial optical solitons in a periodic medium created by linear and nonlinear waveguides. First we consider a single layer with a cubic nonlinear response (a nonlinear slab waveguide) embedded in a periodic layered linear medium and describe nonlinear localized modes (guided waves and Bragg-like localized gap modes) and their stability. Then we study modulational instability as well as the existence and stability of discrete spatial solitons in a periodic array of identical nonlinear layers, a one-dimensional model of nonlinear photonic crystals. We emphasize both similarities to and differences from the models described by the discrete nonlinear Schrödinger equation, which is derived in the tight-binding approximation, and the coupled-mode theory, which is valid for shallow periodic modulations.

© 2002 Optical Society of America

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  1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
    [CrossRef] [PubMed]
  2. Yu. S. Kivshar, “Self-localization in arrays of defocusing wave-guides,” Opt. Lett. 18, 1147–1149 (1993).
    [CrossRef]
  3. W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
    [CrossRef]
  4. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
    [CrossRef]
  5. S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
    [CrossRef]
  6. F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.
  7. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
    [CrossRef]
  8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
    [CrossRef] [PubMed]
  9. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
    [CrossRef]
  10. R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
    [CrossRef] [PubMed]
  11. Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].
  12. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  13. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [CrossRef] [PubMed]
  14. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. XXXIII, pp. 203–260.
  15. A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001).
    [CrossRef] [PubMed]
  16. S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
    [CrossRef] [PubMed]
  17. A. A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” August6, 2001, arXiv.org . e-print archive; Phys. Rev. E (to be published).
  18. I. V. Gerasimchuk and A. S. Kovalev, “Localization of nonlinear waves in layered media,” Fiz. Nizk. Temp. 26, 799–809 (2000) [Low Temp. Phys. 26, 586–593 (2000)].
    [CrossRef]
  19. P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
    [CrossRef]
  20. A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
    [CrossRef]
  21. C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
    [CrossRef]
  22. H. Grebel and W. Zhong, “Holographic interconnects—transverse Bragg wave-guides,” Opt. Lett. 18, 1123–1125 (1993).
    [CrossRef]
  23. R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear structures,” Opt. Lett. 18, 1612–1614 (1993).
    [CrossRef] [PubMed]
  24. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
    [CrossRef]
  25. S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
    [CrossRef]
  26. B. Luther-Davies and G. I. Stegeman, “Materials for spatial solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 19–35.
  27. E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
    [CrossRef]
  28. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).
  29. A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
    [CrossRef]
  30. E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
    [CrossRef]
  31. R. L. Pego and M. I. Weinstein, “Evans’ function, Melnikov’s integral, and solitary wave instabilities,” in Differential Equations with Applications to Mathematical Physics, Vol. 192 of Mathematics in Science and Engineering, W. F. Ames, E. M. Harrell, and J. V. Herod, eds. (Academic, Boston, Mass., 1993), pp. 273–286.
  32. N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)].
    [CrossRef]
  33. Yu. S. Kivshar and A. A. Sukhorukov, “Stability of spatial optical solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 211–245.
  34. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
    [CrossRef]
  35. J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
    [CrossRef]
  36. J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
    [CrossRef]
  37. B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603-4 (2001).
    [CrossRef]
  38. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).
  39. C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
    [CrossRef]
  40. Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
    [CrossRef] [PubMed]
  41. Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
    [CrossRef]
  42. Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
    [CrossRef]
  43. S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
    [CrossRef]
  44. P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
    [CrossRef]
  45. T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
    [CrossRef]
  46. Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
    [CrossRef]
  47. Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
    [CrossRef]
  48. F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
    [CrossRef]

2001 (10)

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603-4 (2001).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
[CrossRef]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
[CrossRef]

2000 (3)

I. V. Gerasimchuk and A. S. Kovalev, “Localization of nonlinear waves in layered media,” Fiz. Nizk. Temp. 26, 799–809 (2000) [Low Temp. Phys. 26, 586–593 (2000)].
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

1999 (2)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

1998 (6)

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
[CrossRef]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
[CrossRef]

1997 (1)

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

1996 (3)

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

1995 (1)

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

1994 (1)

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

1993 (4)

1992 (2)

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

1989 (1)

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

1988 (1)

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1981 (1)

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

1977 (1)

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)].
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Aitchison, J. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Bang, O.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Barra, F.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Bishop, A. R.

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
[CrossRef]

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

Botez, D.

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Bronski, J. C.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

Busch, K.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

Cai, D.

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Campbell, D. K.

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Carr, L. D.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

Carretero-Gonzalez, R.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

Champneys, A. R.

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Chan, C. T.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Cho, A. Y.

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

Christodoulides, D. N.

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

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

Chubykalo, O. A.

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Darmanyan, S.

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998).
[CrossRef]

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

Deconinck, B.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

Gaspard, P.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Gerasimchuk, I. V.

I. V. Gerasimchuk and A. S. Kovalev, “Localization of nonlinear waves in layered media,” Fiz. Nizk. Temp. 26, 799–809 (2000) [Low Temp. Phys. 26, 586–593 (2000)].
[CrossRef]

Grebel, H.

Joseph, R. I.

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

D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13, 794–796 (1988).
[CrossRef] [PubMed]

Kapitula, T.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
[CrossRef]

Kevrekidis, P. G.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
[CrossRef]

Kivshar, Yu. S.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Yu. S. Kivshar, “Self-localization in arrays of defocusing wave-guides,” Opt. Lett. 18, 1147–1149 (1993).
[CrossRef]

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

Yu. S. Kivshar and A. A. Sukhorukov, “Stability of spatial optical solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 211–245.

A. A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” August6, 2001, arXiv.org . e-print archive; Phys. Rev. E (to be published).

Kobyakov, A.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
[CrossRef]

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)].
[CrossRef]

Kovalev, A. S.

I. V. Gerasimchuk and A. S. Kovalev, “Localization of nonlinear waves in layered media,” Fiz. Nizk. Temp. 26, 799–809 (2000) [Low Temp. Phys. 26, 586–593 (2000)].
[CrossRef]

Krolikowski, W.

W. Krolikowski and Yu. S. Kivshar, “Soliton-based optical switching in waveguide arrays,” J. Opt. Soc. Am. B 13, 876–887 (1996).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

Kutz, J. N.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

Lan, S.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Lederer, F.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.

Leine, L.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Li, Q. M.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Lidorikis, E.

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

B. Luther-Davies and G. I. Stegeman, “Materials for spatial solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 19–35.

Malomed, B. A.

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
[CrossRef]

Mann, M.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mingaleev, S. F.

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

Morandotti, R.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Nabiev, R. F.

Nishikawa, S.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Niu, Q.

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603-4 (2001).
[CrossRef]

Pego, R. L.

R. L. Pego and M. I. Weinstein, “Evans’ function, Melnikov’s integral, and solitary wave instabilities,” in Differential Equations with Applications to Mathematical Physics, Vol. 192 of Mathematics in Science and Engineering, W. F. Ames, E. M. Harrell, and J. V. Herod, eds. (Academic, Boston, Mass., 1993), pp. 273–286.

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Peyrard, M.

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

Promislow, K.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

Rasmussen, K. O.

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
[CrossRef]

Rica, S.

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

Ryzhov, Yu. N.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Schmidt, E.

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

Silberberg, Y.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

Sipe, J. E.

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

Smerzi, A.

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001).
[CrossRef] [PubMed]

Sorel, M.

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

Sotin, V. E.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

Soukoulis, C. M.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

Stegeman, G. I.

B. Luther-Davies and G. I. Stegeman, “Materials for spatial solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 19–35.

Sukhorukov, A. A.

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

Yu. S. Kivshar and A. A. Sukhorukov, “Stability of spatial optical solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 211–245.

A. A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” August6, 2001, arXiv.org . e-print archive; Phys. Rev. E (to be published).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Trillo, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Trombettoni, A.

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001).
[CrossRef] [PubMed]

Trutschel, U.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Vakhitov, N. G.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)].
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

Voloshchenko, Yu. I.

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

Wachter, C.

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

Wada, O.

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

Weinstein, M. I.

R. L. Pego and M. I. Weinstein, “Evans’ function, Melnikov’s integral, and solitary wave instabilities,” in Differential Equations with Applications to Mathematical Physics, Vol. 192 of Mathematics in Science and Engineering, W. F. Ames, E. M. Harrell, and J. V. Herod, eds. (Academic, Boston, Mass., 1993), pp. 273–286.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

Wu, B.

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603-4 (2001).
[CrossRef]

Yariv, A.

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Yeh, P.

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

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Zhong, W.

Appl. Phys. Lett. (4)

A. Y. Cho, A. Yariv, and P. Yeh, “Observation of confined propagation in Bragg waveguides,” Appl. Phys. Lett. 30, 471–472 (1977).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, and C. M. Bowden, “Thin-film nonlinear-optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

S. Lan, S. Nishikawa, and O. Wada, “Leveraging deep photonic band gaps in photonic crystal impurity bands,” Appl. Phys. Lett. 78, 2101–2103 (2001).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999).
[CrossRef]

Fiz. Nizk. Temp. (1)

I. V. Gerasimchuk and A. S. Kovalev, “Localization of nonlinear waves in layered media,” Fiz. Nizk. Temp. 26, 799–809 (2000) [Low Temp. Phys. 26, 586–593 (2000)].
[CrossRef]

Izv. Vyssh. Uchebn. Zaved., Radiofiz. (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in the medium with nonlinearity saturation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, 1020–1028 (1973) [Radiophys. Quantum Electron. 16, 783–789 (1973)].
[CrossRef]

J. Appl. Phys. (1)

C. Wachter, F. Lederer, L. Leine, U. Trutschel, and M. Mann, “Nonlinear Bragg reflection wave-guide,” J. Appl. Phys. 71, 3688–3692 (1992).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

P. Yeh and A. Yariv, “Bragg reflection waveguides,” Opt. Commun. 19, 427–430 (1976).
[CrossRef]

Opt. Lett. (4)

Phys. Rep. (1)

Yu. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Phys. Rev. A (2)

Yu. S. Kivshar and M. Peyrard, “Modulational instabilities in discrete lattices,” Phys. Rev. A 46, 3198–3205 (1992).
[CrossRef] [PubMed]

B. Wu and Q. Niu, “Landau and dynamical instabilities of the superflow of Bose–Einstein condensates in optical lattices,” Phys. Rev. A 64, 061603-4 (2001).
[CrossRef]

Phys. Rev. B (3)

E. Lidorikis, Q. M. Li, and C. M. Soukoulis, “Wave propagation in nonlinear multilayer structures,” Phys. Rev. B 54, 10249–10252 (1996).
[CrossRef]

E. Lidorikis, K. Busch, Q. M. Li, C. T. Chan, and C. M. Soukoulis, “Optical nonlinear response of a single nonlinear dielectric layer sandwiched between two linear dielectric structures,” Phys. Rev. B 56, 15090–15099 (1997).
[CrossRef]

Yu. S. Kivshar, A. R. Champneys, D. Cai, and A. R. Bishop, “Multiple states of intrinsic localized modes,” Phys. Rev. B 58, 5423–5428 (1998).
[CrossRef]

Phys. Rev. E (10)

Yu. S. Kivshar and D. K. Campbell, “Peierls–Nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E 48, 3077–3081 (1993).
[CrossRef]

Yu. S. Kivshar, W. Krolikowski, and O. A. Chubykalo, “Dark solitons in discrete lattices,” Phys. Rev. E 50, 5020–5032 (1994).
[CrossRef]

F. Barra, P. Gaspard, and S. Rica, “Nonlinear Schrödinger flow in a periodic potential,” Phys. Rev. E 61, 5852–5863 (2000).
[CrossRef]

P. G. Kevrekidis, A. R. Bishop, and K. O. Rasmussen, “Twisted localized modes,” Phys. Rev. E 63, 036603-6 (2001).
[CrossRef]

T. Kapitula, P. G. Kevrekidis, and B. A. Malomed, “Stability of multiple pulses in discrete systems,” Phys. Rev. E 63, 036604-8 (2001).
[CrossRef]

A. A. Sukhorukov, Yu. S. Kivshar, O. Bang, and C. M. Soukoulis, “Parametric localized modes in quadratic nonlinear photonic structures,” Phys. Rev. E 63, 016615–9 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612-11 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose–Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615-9 (2001).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Phys. Rev. E 53, 1172–1189 (1996).
[CrossRef]

S. Darmanyan, A. Kobyakov, E. Schmidt, and F. Lederer, “Strongly localized vectorial modes in nonlinear waveguide arrays,” Phys. Rev. E 57, 3520–3530 (1998).
[CrossRef]

Phys. Rev. Lett. (8)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. 81, 3383–3386 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Phys. Rev. Lett. 83, 2726–2729 (1999).
[CrossRef]

R. Morandotti, H. S. Eisenberg, Y. Silberberg, M. Sorel, and J. S. Aitchison, “Self-focusing and defocusing in waveguide arrays,” Phys. Rev. Lett. 86, 3296–3299 (2001).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

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

A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2353–2356 (2001).
[CrossRef] [PubMed]

S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

S. Darmanyan, A. Kobyakov, and F. Lederer, “Stability of strongly localized excitations in discrete media with cubic nonlinearity,” Zh. Eksp. Teor. Fiz. 113, 1253–1260 (1998) [JETP 86, 682–686 (1998)].
[CrossRef]

Zh. Tekh. Fiz. (1)

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) [Tech. Phys. 26, 541–544 (1981)].

Other (8)

F. Lederer, S. Darmanyan, and A. Kobyakov, “Discrete solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 269–292.

A. A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” August6, 2001, arXiv.org . e-print archive; Phys. Rev. E (to be published).

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1992).

R. L. Pego and M. I. Weinstein, “Evans’ function, Melnikov’s integral, and solitary wave instabilities,” in Differential Equations with Applications to Mathematical Physics, Vol. 192 of Mathematics in Science and Engineering, W. F. Ames, E. M. Harrell, and J. V. Herod, eds. (Academic, Boston, Mass., 1993), pp. 273–286.

Yu. S. Kivshar and A. A. Sukhorukov, “Stability of spatial optical solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 211–245.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1959).

B. Luther-Davies and G. I. Stegeman, “Materials for spatial solitons,” in Spatial Optical Solitons, S. Trillo and W. E. Torruellas, eds. (Springer-Verlag, New York, 2001), pp. 19–35.

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Figures (13)

Fig. 1
Fig. 1

Characteristic dependence of ζ on β for localized states. Waveguiding (white areas, ζ>0) and antiwaveguiding (dotted area, ζ<0) localization regimes inside the band gaps are shown. The lattice parameters are h=1 and ν(x)=0 for n-1/2<x/h<n and ν(x)=30 for n<x/h<n+1/2, where n is an integer, α=0.5, and γ=1.

Fig. 2
Fig. 2

Top, power versus propagation constant for the nonlinear localized states: solid curves, stable; dashed curve, unstable; dotted curve, oscillatory unstable. Middle, real (dotted curves), and imaginary (solid curves) parts of the eigenvalues associated with wave instability. Waveguiding (white areas) and antiwaveguiding (dotted area) localization regimes inside the band gaps are shown. Bottom, localized states that correspond to points marked a and b at the top. The lattice parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Example of the resonance that occurs between an internal mode of the localized state and a bandgap edge, leading to oscillatory instability.

Fig. 4
Fig. 4

Top, power versus propagation constant. (a)–(c) Localized mode profiles for α=-5 and γ=+1. Notation is the same as in Fig. 2.

Fig. 5
Fig. 5

Characteristic dependencies of parameters η and ξ on propagation constant β. White areas mark bandgaps. The lattice parameters are h=0.5 and α=10.

Fig. 6
Fig. 6

Development of modulational instability in a self-focusing medium for a slightly perturbed unstaggered BW solution with I00.44. The lattice parameters are α=3, h=0.5, and γ=+1.

Fig. 7
Fig. 7

Modulationally unstable staggered BW modes (shaded area) in a self-focusing medium, shown as intensity (I0) versus lattice parameter α (at h=0.5 and γ=+1). Dashed curve, analytical approximation (16) for the low-intensity instability threshold.

Fig. 8
Fig. 8

Unstable modulation frequencies (shaded area) versus intensity I0 of the staggered BW modes (α=3 and h=0.5). Dark, solid curve, instability frequency with the largest growth rate.

Fig. 9
Fig. 9

Development of the instability-induced period-doubling modulations. The initial profile corresponds to a slightly perturbed staggered solution with I029.87. The parameters are the same as in Fig. 8.

Fig. 10
Fig. 10

Top, power versus propagation constant for odd (dark curve) and even (light curve) localized modes in a self-focusing (γ=+1) regime: solid curve stable; dashed curve, unstable. Bottom, profiles of the localized modes that correspond to marked points a and b at the top. The lattice parameters are the same as in Fig. 5.

Fig. 11
Fig. 11

Top, power versus propagation constant in the self-defocusing (γ=-1) regime. Notation is the same as in Fig. 10; dotted curve, oscillatory unstable modes.

Fig. 12
Fig. 12

Top, complementary power versus propagation constant for odd (dark curve) and even (lighter curve) dark localized solitons in a self-focusing (γ=+1) regime. Notation is the same as in Fig. 10.

Fig. 13
Fig. 13

Top, complementary power versus propagation constant in the self-defocusing (γ=-1) regime. Notation is the same as in Fig. 12.

Equations (29)

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i EZ+D 2EX2+(X)E+g(X)|E|2E=0,
i ψz+2ψx2+F(I; x)ψ=0,
P=-+|ψ(x, z)|2dx
ψ(x, z)=u(x;β)exp(iβz),
-βu+d2udx2+F(I; x)u=0.
F(I; x)=ν(x)+δ(x)G(I),
ub(x;β)=a(x;β)exp[-μ(x;β)x]+b(x;β)exp[+μ(x;β)x],
T(x;β)1r(x;β)=τ(β)1r(x;β),
G0G(I0)=ζ(β),
ζ±=μ± (1-r±)(1+r±);
ψ(x, t)=[u(x)+v(x)exp(iΓz)+w*(x)exp(-iΓ*z)]exp(iβz),
-(β+Γ)v+d2vdx2+ν(x)v+δ(x)[G1v
+(G1-G0)w]=0,
-(β-Γ)w+d2wdx2+ν(x)w+δ(x)[G1w
+(G1-G0)v]=0,
Y(Γ)=[G1-ζ(β+Γ)][G1-ζ(β-Γ)]-(G1-G0)2.
F(I; x)=n(α+γI)δ(x-hn),
ηUn+(Un-1+Un+1)+χ|Un|2Un=0.
η=-2 cosh(hμ)+αξ,
ξ=sinh(hμ)/μ,μ=β.
I0(β)=-[2 cos K+η(β)]γξ(β).
[η(β+Γ)+2γξ(β+Γ)I0+2 cos(q+K)]×[η(β-Γ)+2γξ(β-Γ)I0+2 cos(q-K)]
=γ2ξ(β+Γ)ξ(β-Γ)I02.
γI0(cr)α+22hα3/2/π+O(α2).
-(β+Γ)v+d2vdx2+n[(α+2γun2)v+γun2w]
×δ(x-hn)=0,
-(β-Γ)w+d2wdx2+n[(α+2γun2)w+γun2v]
×δ(x-hn)=0,
Pc=limn+-nh+nh[|u(x+2nh)|2-|u(x)|2]dx,

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