Abstract

We introduce a system of nonlinear coupled-mode equations (CMEs) for Bragg gratings (BGs) where the Bragg reflectivity periodically switches off and on as a function of the evolution variable. The model may be realized in a planar waveguide with the Kerr nonlinearity, where the grating is represented by an array of parallel dashed lines (grooves), aligned with the propagation direction. In the temporal domain, a similar system can be derived for matter waves trapped in a rocking optical lattice. Using systematic simulations, we construct families of gap solitons (GSs) in this system, starting with inputs provided by exact GS solutions in the averaged version of the CMEs. Four different regimes of the dynamical behavior are identified: fully stable, weakly unstable, moderately unstable, and completely unstable solitons. The analysis is reported for both quiescent and moving solitons (in fact, they correspond to untilted and tilted beams in the spatial domain). Weakly and moderately unstable GSs spontaneously turn into persistent breathers (the moderate instability entails a small spontaneous change of the breather’s velocity). Stability regions for the solitons and breathers are identified in the parameter space. Collisions between stably moving solitons and breathers always appear to be elastic.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  44. 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]
  45. D. R. Neill and J. Atai, “Collision dynamics of gap solitons in Kerr media,” Phys. Lett. A 353, 416–421 (2006).
    [CrossRef]
  46. D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A, Pure Appl. Opt. 10, 085105 (2008).
    [CrossRef]

2009 (2)

A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, “Exploring dynamic localization with a Bose–Einstein condensate,” Phys. Rev. A 79, 013611 (2009).
[CrossRef]

T. Mayteevarunyoo and B. A. Malomed, “Gap solitons in rocking optical lattices and waveguides with undulating gratings,” Phys. Rev. A 80, 013827 (2009).
[CrossRef]

2008 (5)

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101, 143602 (2008).
[CrossRef] [PubMed]

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

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[CrossRef]

K. Levy and B. A. Malomed, “Stability and collisions of traveling solitons in Bragg-grating superstructures,” J. Opt. Soc. Am. B 25, 302–309 (2008).
[CrossRef]

T. Mayteevarunyoo and B. A. Malomed, “Gap solitons in grating superstructures,” Opt. Express 16, 7767–7777 (2008).
[CrossRef] [PubMed]

2007 (1)

H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403 (2007).
[CrossRef]

2006 (3)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[CrossRef]

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

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

2005 (3)

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[CrossRef]

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

2004 (3)

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

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

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and splitting of gap solitons in apodized Bragg gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[CrossRef]

2003 (5)

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

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

P. G. Kevrekidis, B. A. Malomed, and Z. Musslimani, “Discrete gap solitons in a diffraction-managed waveguide array,” Eur. Phys. J. D 23, 421–436 (2003).
[CrossRef]

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

2002 (2)

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

J. Atai and B. A. Malomed, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140–148 (2002).
[CrossRef]

2001 (1)

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905–3909 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (3)

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

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

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

1996 (1)

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

1994 (1)

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

1993 (2)

1989 (2)

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

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

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]

1986 (1)

P. St. J. Russell, “Optical superlattices for modulation and deflection of light,” J. Appl. Phys. 59, 3344–3355 (1986).
[CrossRef]

1981 (1)

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with large group retardation,” Zh. Tekh. Fiz. 51, 902 (1981); Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin,[Sov. Phys. Tech. Phys. 26, 541 (1981)].

Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in nonlinear, periodically modulated media with large group retardation,” Zh. Tekh. Fiz. 51, 902 (1981); Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin,[Sov. Phys. Tech. Phys. 26, 541 (1981)].

1979 (1)

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]

1977 (1)

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634–3646 (1977).
[CrossRef]

Aceves, A. B.

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

Agrawal, G. P.

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

Ahuja, A.

Aitchison, J. S.

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

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

Amann, A.

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[CrossRef]

Arimondo, E.

A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, “Exploring dynamic localization with a Bose–Einstein condensate,” Phys. Rev. A 79, 013611 (2009).
[CrossRef]

H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403 (2007).
[CrossRef]

Atai, J.

D. R. Neill, J. Atai, and B. A. Malomed, “Dynamics and collisions of moving solitons in Bragg gratings with dispersive reflectivity,” J. Opt. A, Pure Appl. Opt. 10, 085105 (2008).
[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, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140–148 (2002).
[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]

Belmonte, M.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

Biancalana, F.

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[CrossRef]

Botez, D.

Broderick, N. G. R.

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634–3646 (1977).
[CrossRef]

Carmon, T.

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

Champneys, A. R.

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
[CrossRef]

Chen, F.

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]

Christodoulides, D. N.

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

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

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

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Slowdown and splitting of gap solitons in apodized Bragg gratings,” J. Mod. Opt. 51, 2141–2158 (2004).
[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]

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]

Ciampini, D.

A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, “Exploring dynamic localization with a Bose–Einstein condensate,” Phys. Rev. A 79, 013611 (2009).
[CrossRef]

H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403 (2007).
[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]

De Rossi, A.

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

de Sterke, C. M.

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

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

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

N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E 55, 3634–3646 (1977).
[CrossRef]

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

Della Valle, G.

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

Dreisow, F.

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Appl. Phys. Lett. (1)

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Eur. Phys. J. D (1)

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Europhys. Lett. (1)

K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. 74, 1006–1012 (2006).
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J. Appl. Phys. (2)

R. Shimada, T. Koda, T. Ueta, and K. Ohtaka, “Strong localization of Bloch photons in dual-periodic dielectric multilayer structures,” J. Appl. Phys. 90, 3905–3909 (2001).
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J. Lightwave Technol. (1)

J. Mod. Opt. (1)

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J. Opt. A, Pure Appl. Opt. (1)

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

Nat. Phys. (1)

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

Nature (1)

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

Opt. Express (2)

Opt. Lett. (3)

Phys. Lett. A (3)

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[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, “Spatial solitons in a medium composed of self-focusing and self-defocusing layers,” Phys. Lett. A 298, 140–148 (2002).
[CrossRef]

Phys. Rev. A (5)

A. Eckardt, M. Holthaus, H. Lignier, A. Zenesini, D. Ciampini, O. Morsch, and E. Arimondo, “Exploring dynamic localization with a Bose–Einstein condensate,” Phys. Rev. A 79, 013611 (2009).
[CrossRef]

T. Mayteevarunyoo and B. A. Malomed, “Gap solitons in rocking optical lattices and waveguides with undulating gratings,” Phys. Rev. A 80, 013827 (2009).
[CrossRef]

F. Biancalana, A. Amann, and E. P. O’Reilly, “Gap solitons in spatiotemporal photonic crystals,” Phys. Rev. A 77, 011801(R) (2008).
[CrossRef]

P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A 71, 032612 (2005).
[CrossRef]

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[CrossRef]

Phys. Rev. E (5)

D. Janner, G. Galzerano, G. Della Valle, P. Laporta, S. Longhi, and M. Belmonte, “Slow light in periodic superstructure Bragg gratings,” Phys. Rev. E 72, 056605 (2005).
[CrossRef]

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[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).
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[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]

Phys. Rev. Lett. (11)

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S. Longhi, “Decay control via discrete-to-continuum coupling modulation in an optical waveguide system,” Phys. Rev. Lett. 101, 143602 (2008).
[CrossRef] [PubMed]

H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zenesini, O. Morsch, and E. Arimondo, “Dynamical control of matter-wave tunneling in periodic potentials,” Phys. Rev. Lett. 99, 220403 (2007).
[CrossRef]

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[CrossRef]

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D. Neshev, A. A. Sukhorukov, B. Hanna, W. Królikowski, and Yu. S. Kivshar, “Controlled generation and steering of spatial gap solitons,” Phys. Rev. Lett. 93, 083905 (2004).
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D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet–Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
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Zh. Tekh. Fiz. (1)

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Other (4)

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

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

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

B. A. Malomed, Soliton Management in Periodic Systems (Springer, 2006).

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Figures (15)

Fig. 1
Fig. 1

Typical example of a stable nearly stationary quiescent soliton for κ = 1.01 , δ = π 8 , and Z map = 4.2 . In this figure and in Figs. 2, 3, 4 , panels (a) and (b) display the evolution of the u and v components, respectively. Panel (c) shows the evolution of the soliton’s power.

Fig. 2
Fig. 2

Typical example of a stable breather generated by the evolution of a weakly unstable quiescent soliton, for κ = 1.5 , δ = π 10 , and Z map = 1 .

Fig. 3
Fig. 3

Evolution of a moderately unstable initially quiescent soliton, which spontaneously transforms itself into a moving breather, at κ = 1.15 , δ = π 8 , and Z map = 2.5 . Panel (c) displays a detailed picture of the soliton’s motion by means of contour plots of | u ( x , z ) | 2 (for | v ( x , z ) | 2 the picture is virtually identical).

Fig. 4
Fig. 4

Example of the decay of a completely unstable quiescent soliton, for δ = π 8 , κ = 1.1 , and Z map = 3.0 .

Fig. 5
Fig. 5

Stability borders for the (initially) quiescent solitons in the plane of ( δ , Z map ) for fixed κ = 1.01 (a), 1.1 (b), 1.3 (c) and 1.7 (b). In panel (a), which corresponds to very small κ 1 = 0.01 , the diagram does not have a region of completely unstable (decaying) solitons. The vertical lines at δ π 2 represent the stability border in the standard model with the uniform reflectivity, κ ( x ) 1 [10, 11, 12]; see the text.

Fig. 6
Fig. 6

Stability borders for the (initially) quiescent solitons in the plane of ( κ 1 , Z map ) for fixed δ = π 10 .

Fig. 7
Fig. 7

Example of a stable moving soliton generated by the initial condition (4) with δ = π 30 , κ = 1.2 , and velocity parameter c = 0.6 in initial conditions (4) (the actual average velocity produced by the simulations is c ¯ 0.667 ). The management period is Z map = 0.6 . Here and in Figs. 8, 9, coordinate y is defined as per Eq. (10), with appropriate values of c ¯ .

Fig. 8
Fig. 8

Example of a stable moving breather with δ = π 30 , κ = 1.2 , c = 0.6 , and Z map = 1.2 . The average velocity produced by the simulations is c ¯ 0.658 .

Fig. 9
Fig. 9

Example of moderately unstable moving breather with δ = π 10 , c = 0.6 , Z map = 1.5 , and κ = 1.2 . The average velocity produced by the simulations is c ¯ 0.538 (note that c ¯ < c in this case, on the contrary to c ¯ > c in the cases displayed in Figs. 7, 9).

Fig. 10
Fig. 10

Stability borders for moving solitons with velocities c = 0.1 and c = 0.9 , at ( κ 1 ) = 0.01 .

Fig. 11
Fig. 11

Stability borders for the moving solitons with κ 1 = 0.2 : (a) c = 0.2 , (b) c = 0.6 .

Fig. 12
Fig. 12

Residual velocity (defined in the text) as functions of κ 1 and Z map , at fixed Z map = 1 and κ = 1.2 , respectively. In both cases, δ = π 10 .

Fig. 13
Fig. 13

Collision between stable solitons moving with velocities c = ± 0.6 , the other parameters being κ 1 = 0.2 , Z map = 0.5 , and δ = π 10 .

Fig. 14
Fig. 14

Collision between two stable moving solitons at κ 1 = 0.2 and Z map = 0.5 . The initial parameters of the left and right solitons are, respectively, c = 0.6 , δ = π 10 , and c = 0.5 , δ = π 30 .

Fig. 15
Fig. 15

Collision between stable breathers moving with velocities c = ± 0.6 , the other parameters being κ 1 = 0.2 , Z map = 1.2 , and δ = π 30 .

Equations (15)

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i u t + i u x + ( | u | 2 + 2 | v | 2 ) u + κ v = 0 ,
i v t i v x + ( | v | 2 + 2 | u | 2 ) v + κ u = 0 .
i u z + i u x + ( | u | 2 + 2 | v | 2 ) u + κ ( z ) v = 0 ,
i v z i v x + ( | v | 2 + 2 | u | 2 ) v + κ ( z ) u = 0 ,
κ ( Z map ) = { κ 0 < z < Z on 0 Z on < z < Z on + Z off } .
u 0 ( x ) = A 1 ( sin δ ) W exp ( i σ ) sech ( θ i δ 2 ) ,
v 0 ( x ) = A ( sin δ ) W exp ( i σ ) sech ( θ + i δ 2 ) ,
A = [ ( 1 c ) ( 1 + c ) ] 1 4 , γ = ( 1 c 2 ) 1 2 ,
θ = γ ( sin δ ) ( x c z ) , σ = γ ( cos δ ) ( c x z ) ,
W = 1 c 2 3 c 2 [ exp ( 2 θ ) + exp ( i δ ) exp ( 2 θ ) + exp ( i δ ) ] 2 c 3 c 2 .
E + [ | u ( x ) | 2 + | v ( x ) | 2 ] d x = 4 θ ( 1 c 2 ) ( 3 c 2 ) ,
P i + ( u x * u + v x * v ) d x = 4 c 1 c 2 × [ 7 c 2 ( 3 c 2 ) 2 ( sin δ δ cos δ ) + δ cos δ 3 c 2 ] .
y = x c ¯ z ,
i u z + i ( 1 c ¯ ) u y + ( | u | 2 + 2 | v | 2 ) u + κ ( z ) v = 0 ,
i v z i ( 1 + c ¯ ) v x + ( | v | 2 + 2 | u | 2 ) v + κ ( z ) u = 0 ,

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