Abstract

The existence, stability, and collisions of moving solitons in Bragg gratings (BGs) in a cubic-quintic nonlinear medium are investigated. Two disjoint families of solitons that are separated by a border are identified. One family (Type 1) can be considered as the generalization of the moving solitons in BGs written in a cubic nonlinear medium. The other family (Type 2) occurs in regions where quintic nonlinearity dominates. Through systematic numerical stability analysis, the stability regions in the plane of quintic nonlinearity versus frequency have been determined. It is found that the stability regions are dependent on the velocity of solitons. The collisions of counterpropagating solitons have been systematically investigated. Collisions of in-phase Type 1 solitons can result in a variety of outcomes including forming two asymmetrically separating solitons and passing through each other and separating symmetrically with reduced, unchanged, or increased velocities. In certain parameter regions, solitons merge to form a quiescent one. An outcome that has not been reported previously for uniform gratings is the formation of a quiescent soliton and two symmetrically separating solitons. This outcome is found to be more robust than the merger.

© 2013 Optical Society of America

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    [CrossRef]
  40. H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4315 (1997).
    [CrossRef]
  41. 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]
  42. J. Atai and B. A. Malomed, “Families of Bragg-grating solitons in a cubic–quintic medium,” Phys. Lett. A 284, 247–252 (2001).
    [CrossRef]
  43. J. Atai, “Interaction of Bragg grating solitons in a cubic–quintic medium,” J. Opt. B Quantum Semiclass. Opt. 6, S177–S181 (2004).
    [CrossRef]
  44. G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
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    [CrossRef]
  47. S. Dasanayaka and J. Atai, “Interactions of solitons in Bragg gratings with dispersive reflectivity in a cubic-quintic medium,” Phys. Rev. E 84, 026613 (2011).
    [CrossRef]
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    [CrossRef]
  49. E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, “Coherent interactions of dissipative spatial solitons,” Opt. Lett. 29, 283–285 (2004).
    [CrossRef]
  50. J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
    [CrossRef]
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    [CrossRef]
  52. I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
    [CrossRef]

2012

B. H. Baratali, and J. Atai, “Gap solitons in dual-core Bragg gratings with dispersive reflectivity,” J. Opt. 14, 065202 (2012).
[CrossRef]

2011

S. Dasanayaka and J. Atai, “Interactions of solitons in Bragg gratings with dispersive reflectivity in a cubic-quintic medium,” Phys. Rev. E 84, 026613 (2011).
[CrossRef]

2010

S. Dasanayaka and J. Atai, “Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity,” Phys. Lett. A 375, 225–229 (2010).
[CrossRef]

2008

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A 10, 085105 (2008).
[CrossRef]

2007

D. R. Neill, and J. Atai, “Gap solitons in a hollow optical fiber in the normal dispersion regime,” Phys. Lett. A 367, 73–82 (2007).
[CrossRef]

Y. P. Shapira and M. Horowitz, “Optical AND gate based on soliton interaction in a fiber Bragg grating,” Opt. Lett. 32, 1211–1213 (2007).
[CrossRef]

2006

J. Cuevas and J. C. Eilbeck, “Discrete soliton collisions in a waveguide array with saturable nonlinearity,” Phys. Lett. A 358, 15–20 (2006).
[CrossRef]

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

J. Atai, B. A. Malomed, and I. M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber,” Opt. Commun. 265, 342–348 (2006).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

2005

J. Atai, and B. A. Malomed, “Gap solitons in Bragg gratings with dispersive reflectivity,” Phys. Lett. A 342, 404–412 (2005).
[CrossRef]

I. M. Merhasin and B. A. Malomed, “Gap solitons in a model of a hollow optical fiber,” Opt. Lett. 30, 1105–1107 (2005).
[CrossRef]

2004

E. A. Ultanir, G. I. Stegeman, C. H. Lange, and F. Lederer, “Coherent interactions of dissipative spatial solitons,” Opt. Lett. 29, 283–285 (2004).
[CrossRef]

D. V. Skryabin, “Coupled core-surface solitons in photonic crystal fibers,” Opt. Express 12, 4841–4846 (2004).
[CrossRef]

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

J. Atai, “Interaction of Bragg grating solitons in a cubic–quintic medium,” J. Opt. B Quantum Semiclass. Opt. 6, S177–S181 (2004).
[CrossRef]

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

2003

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[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]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

2002

2001

J. Atai and B. A. Malomed, “Families of Bragg-grating solitons in a cubic–quintic medium,” Phys. Lett. A 284, 247–252 (2001).
[CrossRef]

J. Atai, and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
[CrossRef]

2000

J. Atai, and B. A. Malomed, “Bragg-grating solitons in a semilinear dual-core system,” Phys. Rev. E 62, 8713–8718 (2000).
[CrossRef]

1998

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]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685–1692 (1998).
[CrossRef]

D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
[CrossRef]

1997

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

W. Królikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Opt. Lett. 22, 369–371 (1997).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4315 (1997).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (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

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[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]

1995

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

S. Radic, N. George, and G. P. Agrawal, “Theory of low-threshold optical switching in nonlinear phase-shifted periodic structures,” J. Opt. Soc. Am. B 12, 671–680 (1995).
[CrossRef]

1994

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]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

1992

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

1991

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

1990

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

1989

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

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

1988

1985

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

1979

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[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.

Aitchison, J. S.

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

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

Anderson, C. J.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

Assanto, G.

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

Atai, J.

B. H. Baratali, and J. Atai, “Gap solitons in dual-core Bragg gratings with dispersive reflectivity,” J. Opt. 14, 065202 (2012).
[CrossRef]

S. Dasanayaka and J. Atai, “Interactions of solitons in Bragg gratings with dispersive reflectivity in a cubic-quintic medium,” Phys. Rev. E 84, 026613 (2011).
[CrossRef]

S. Dasanayaka and J. Atai, “Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity,” Phys. Lett. A 375, 225–229 (2010).
[CrossRef]

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A 10, 085105 (2008).
[CrossRef]

D. R. Neill, and J. Atai, “Gap solitons in a hollow optical fiber in the normal dispersion regime,” Phys. Lett. A 367, 73–82 (2007).
[CrossRef]

J. Atai, B. A. Malomed, and I. M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber,” Opt. Commun. 265, 342–348 (2006).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

J. Atai, and B. A. Malomed, “Gap solitons in Bragg gratings with dispersive reflectivity,” Phys. Lett. A 342, 404–412 (2005).
[CrossRef]

J. Atai, “Interaction of Bragg grating solitons in a cubic–quintic medium,” J. Opt. B Quantum Semiclass. Opt. 6, S177–S181 (2004).
[CrossRef]

J. Atai and B. A. Malomed, “Families of Bragg-grating solitons in a cubic–quintic medium,” Phys. Lett. A 284, 247–252 (2001).
[CrossRef]

J. Atai, and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
[CrossRef]

J. Atai, and B. A. Malomed, “Bragg-grating solitons in a semilinear dual-core system,” Phys. Rev. E 62, 8713–8718 (2000).
[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]

Baratali, B. H.

B. H. Baratali, and J. Atai, “Gap solitons in dual-core Bragg gratings with dispersive reflectivity,” J. Opt. 14, 065202 (2012).
[CrossRef]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Broderick, N. G. R.

Brown, T. G.

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Busch, K.

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

Cavaciuti, A.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[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]

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]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685–1692 (1998).
[CrossRef]

Cole, M. J.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

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]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

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J. Cuevas and J. C. Eilbeck, “Discrete soliton collisions in a waveguide array with saturable nonlinearity,” Phys. Lett. A 358, 15–20 (2006).
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S. Dasanayaka and J. Atai, “Interactions of solitons in Bragg gratings with dispersive reflectivity in a cubic-quintic medium,” Phys. Rev. E 84, 026613 (2011).
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S. Dasanayaka and J. Atai, “Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity,” Phys. Lett. A 375, 225–229 (2010).
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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.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
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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).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Dhosi, G.

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
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H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4315 (1997).
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J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (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]

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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J. Cuevas and J. C. Eilbeck, “Discrete soliton collisions in a waveguide array with saturable nonlinearity,” Phys. Lett. A 358, 15–20 (2006).
[CrossRef]

Frantzeskakis, D. J.

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

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H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
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He, H.

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4315 (1997).
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Hibino, Y.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Hill, P.

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

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Horowitz, M.

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D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
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R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

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I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

Królikowski, W.

Krug, P. A.

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[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]

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
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D. Taverner, N. G. R. Broderick, D. J. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
[CrossRef]

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

Lange, C. H.

LaRochelle, S.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Lederer, F.

Li, D.

Li, Y.

Litchinitser, N. M.

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]

Littler, I. C. M.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Loh, W. H.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

Lu, Z.

Mak, W. C. K.

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]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685–1692 (1998).
[CrossRef]

Malomed, B. A.

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A 10, 085105 (2008).
[CrossRef]

J. Atai, B. A. Malomed, and I. M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber,” Opt. Commun. 265, 342–348 (2006).
[CrossRef]

J. Atai, and B. A. Malomed, “Gap solitons in Bragg gratings with dispersive reflectivity,” Phys. Lett. A 342, 404–412 (2005).
[CrossRef]

I. M. Merhasin and B. A. Malomed, “Gap solitons in a model of a hollow optical fiber,” Opt. Lett. 30, 1105–1107 (2005).
[CrossRef]

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[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]

J. Atai and B. A. Malomed, “Families of Bragg-grating solitons in a cubic–quintic medium,” Phys. Lett. A 284, 247–252 (2001).
[CrossRef]

J. Atai, and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
[CrossRef]

J. Atai, and B. A. Malomed, “Bragg-grating solitons in a semilinear dual-core system,” Phys. Rev. E 62, 8713–8718 (2000).
[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]

W. C. K. Mak, P. L. Chu, and B. A. Malomed, “Solitary waves in coupled nonlinear waveguides with Bragg gratings,” J. Opt. Soc. Am. B 15, 1685–1692 (1998).
[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]

Mandelik, D.

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

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Meier, J.

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

Merhasin, I. M.

J. Atai, B. A. Malomed, and I. M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber,” Opt. Commun. 265, 342–348 (2006).
[CrossRef]

I. M. Merhasin and B. A. Malomed, “Gap solitons in a model of a hollow optical fiber,” Opt. Lett. 30, 1105–1107 (2005).
[CrossRef]

Mizrahi, V.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Mok, J. T.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Morandotti, R.

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

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

Neill, D. R.

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A 10, 085105 (2008).
[CrossRef]

D. R. Neill, and J. Atai, “Gap solitons in a hollow optical fiber in the normal dispersion regime,” Phys. Lett. A 367, 73–82 (2007).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
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Niegemann, J.

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

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P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

Papacharalampous, I. E.

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (2003).
[CrossRef]

Patterson, D. B.

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]

Pelinovsky, D. E.

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]

Pereira, S.

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

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N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

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Richardson, D. J.

Robinson, N.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

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G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

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N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

Shapira, Y. P.

Silberberg, Y.

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

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

Sipe, J. E.

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]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

Skryabin, D. V.

Slusher, R. E.

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

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J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
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S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

Stephens, T.

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

Tasgal, R. S.

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]

Taverner, D.

Tkeshelashvili, L.

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

Trillo, S.

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

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

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G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
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Ultanir, E. A.

Vaninetti, F.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
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J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
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H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Yoffe, G.

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

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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]

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W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
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Zhang, D.

Zhao, L.

Zhu, D.

Appl. Phys. Lett.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, “All-optical switching in a nonlinear periodic-waveguide structure,” Appl. Phys. Lett. 60, 1427–1429 (1992).
[CrossRef]

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

Electron. Lett.

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, “All-optical switching of grating transmission using cross-phase modulation in optical fibres,” Electron. Lett. 26, 1459–1460 (1990).
[CrossRef]

P. A. Krug, T. Stephens, G. Yoffe, F. Ouellette, P. Hill, and G. Dhosi, “Dispersion compensation over 270 km at 10  Gbit/s using an offset-core chirped fibre Bragg grating,” Electron. Lett. 31, 1091–1093 (1995).
[CrossRef]

IEEE Photon. Technol. Lett.

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, “Dispersion compensation over distances in excess of 500 km for 10  Gb/ssystems using chirped fiber gratings,” IEEE Photon. Technol. Lett. 8, 944–946 (1996).
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J. Lightwave Technol.

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]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

J. Mod. Opt.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
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J. Opt.

B. H. Baratali, and J. Atai, “Gap solitons in dual-core Bragg gratings with dispersive reflectivity,” J. Opt. 14, 065202 (2012).
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J. Opt. A

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A 10, 085105 (2008).
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J. Opt. Soc. Am. B

Nat. Phys.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775–780 (2006).
[CrossRef]

Opt. Commun.

J. Atai, B. A. Malomed, and I. M. Merhasin, “Stability and collisions of gap solitons in a model of a hollow optical fiber,” Opt. Commun. 265, 342–348 (2006).
[CrossRef]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Photon. Nanostr. Fundam. Appl.

L. Tkeshelashvili, J. Niegemann, S. Pereira, and K. Busch, “Nonlinear wave interaction in photonic band gap materials,” Photon. Nanostr. Fundam. Appl. 4, 75–88 (2006).
[CrossRef]

Phys. Lett. A

J. Atai, and B. A. Malomed, “Gap solitons in Bragg gratings with dispersive reflectivity,” Phys. Lett. A 342, 404–412 (2005).
[CrossRef]

D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
[CrossRef]

S. Dasanayaka and J. Atai, “Stability of Bragg grating solitons in a cubic–quintic nonlinear medium with dispersive reflectivity,” Phys. Lett. A 375, 225–229 (2010).
[CrossRef]

D. R. Neill, and J. Atai, “Gap solitons in a hollow optical fiber in the normal dispersion regime,” Phys. Lett. A 367, 73–82 (2007).
[CrossRef]

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 284, 247–252 (2001).
[CrossRef]

J. Cuevas and J. C. Eilbeck, “Discrete soliton collisions in a waveguide array with saturable nonlinearity,” Phys. Lett. A 358, 15–20 (2006).
[CrossRef]

Phys. Rev. E

I. E. Papacharalampous, P. G. Kevrekidis, B. A. Malomed, and D. J. Frantzeskakis, “Soliton collisions in the discrete nonlinear Schrödinger equation,” Phys. Rev. E 68, 046604 (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]

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]

J. Atai, and B. A. Malomed, “Bragg-grating solitons in a semilinear dual-core system,” Phys. Rev. E 62, 8713–8718 (2000).
[CrossRef]

J. Atai, and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
[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]

S. Dasanayaka and J. Atai, “Interactions of solitons in Bragg gratings with dispersive reflectivity in a cubic-quintic medium,” Phys. Rev. E 84, 026613 (2011).
[CrossRef]

Phys. Rev. Lett.

J. Meier, G. I. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Nonlinear optical beam interactions in waveguide arrays,” Phys. Rev. Lett. 93, 093903 (2004).
[CrossRef]

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[CrossRef]

H. He and P. D. Drummond, “Ideal soliton environment using parametric band gaps,” Phys. Rev. Lett. 78, 4311–4315 (1997).
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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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Prog. Opt.

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[CrossRef]

Other

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Figures (9)

Fig. 1.
Fig. 1.

Effect of soliton velocity on the location of the boundary between Type 1 and Type 2 regions for several values of frequency. The points are obtained numerically, while the solid curves are defined by Eq. (5).

Fig. 2.
Fig. 2.

Examples of Type 1 and Type 2 solitons and their real and imaginary parts for ω=0.5 and v=0.5. (a) U-component and (b) V-component of Type 1 soliton corresponding to η=0.3. (c) U-component and (d) V-component of Type 2 soliton corresponding to η=0.5.

Fig. 3.
Fig. 3.

Stability diagrams for (a) v=0.1, (b) v=0.3, and (c) v=0.5. The dashed curve is the boundary between Type 1 and Type 2 solitons (see the text).

Fig. 4.
Fig. 4.

Examples of collisions of π-out-of-phase solitons (i.e., Δφ=π). (a) Type 1 solitons with ω=0.2, η=0.1, and v=0.5. (b) Type 2 solitons with ω=0.0, η=0.8, and v=0.5. (c) Type 2 solitons with ω=0.3, η=0.8, and v=0.3.

Fig. 5.
Fig. 5.

Examples of soliton–soliton collisions for in-phase (Δφ=0) Type 1 solitons. (a) Separation with unchanged velocities for ω=0.70, η=0.20, and v=0.50. (b) Increase in velocity for ω=0.25, η=0.13, and v=0.10. (c) Decrease in velocity for ω=0.60, η=0.32, and v=0.30. (d) Asymmetric separation after multiple collisions for ω=0.50, η=0.22, and v=0.10. (e) Merger into a quiescent soliton for ω=0.65, η=0.05, and v=0.10. (f) Generation of three solitons for ω=0.55, η=0.35, and v=0.30. Only |u| is shown.

Fig. 6.
Fig. 6.

Collision outcome diagrams for initial soliton velocities of (a) 0.1, (b) 0.3, and (c) 0.5. The labeled regions are as follows: A, formation of two separating solitons with different velocities; D, destruction; E, quasi-elastic collision resulting in passage of solitons through each other; F, generation of two solitons whose velocities are higher than the original ones; M, merger of solitons to form a single quiescent soliton; S, generation of two solitons whose velocities are lower than the original ones; T, formation of three solitons: a quiescent soliton and two moving solitons propagating in opposite directions. The dashed curve separates Type 1 and Type 2 solitons.

Fig. 7.
Fig. 7.

Variation of the velocities of solitons produced by collisions of in-phase Type 1 solitons corresponding to ω=0.4 and v=±0.3, for the range of η, for which symmetric separation occurs. The solid curve represents the final velocity of the separating solitons. The dashed line denotes the initial velocity of solitons, and the dotted lines indicate region boundaries in Fig. 6(b).

Fig. 8.
Fig. 8.

Effect of initial phase difference on the final velocities of emerging solitons for parameters where in-phase collisions cause (a) merger (ω=0.65, η=0.05, and v=0.10) and (b) three-soliton formation (ω=0.55, η=0.35, and v=0.30). The solid curves represent the final velocities of the emerging solitons. The dashed lines indicate the initial velocities of solitons.

Fig. 9.
Fig. 9.

Collision of solitons with a 10% velocity difference (ω=0.55, η=0.35, v1=0.33, and v2=0.30) for (a) in-phase solitons (Δφ=0°) and (b) solitons with phase difference tuned for three soliton formation (Δφ=51°).

Equations (6)

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iψt+iψx+[12|ψ|2+|ϕ|2]ψη[14|ψ|4+32|ψ|2|ϕ|2+34|ϕ|4]ψ+ϕ=0,iϕtiϕx+[12|ϕ|2+|ψ|2]ϕη[14|ϕ|4+32|ϕ|2|ψ|2+34|ψ|4]ϕ+ψ=0,
iψT+i(1v)ψX+[12|ψ|2+|ϕ|2]ψη[14|ψ|4+32|ψ|2|ϕ|2+34|ϕ|4]ψ+ϕ=0,iϕTi(1+v)ϕX+[12|ϕ|2+|ψ|2]ϕη[14|ϕ|4+32|ϕ|2|ψ|2+34|ψ|4]ϕ+ψ=0.
{ψ(X,T),ϕ(X,T)}={U(X),V(X)}eiΩT.
ω1v2U+i(1v)UX+[12|U|2+|V|2]Uη[14|U|4+32|U|2|V|2+34|V|4]U+V=0,ω1v2Vi(1+v)VX+[12|V|2+|U|2]Vη[14|V|4+32|V|2|U|2+34|U|4]V+U=0.
η27160(1ω)(11v2).
ψ(x,0)=Uv(x+Δx2)+Uv(xΔx2)eiΔφ,ϕ(x,0)=Vv(x+Δx2)+Vv(xΔx2)eiΔφ,

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