Abstract

Fundamental properties of solitons in the recently introduced split-step model (SSM) are investigated. The SSM is a system that consists of periodically alternating dispersive and nonlinear segments, a period being of the same order of magnitude as the soliton’s dispersion length. The model including fiber loss and gain can always be reduced to its lossless version. First, we develop a variational approximation that makes it possible to explain the existence of SSM solitons that were originally found solely in numerical form. Overall dynamic behavior of a SSM is described by a phase diagram that identifies an established state (stationary soliton, breather with long-period oscillations, splitting into several pulses, or decay into radiation) depending on the amplitude and the width of the initial pulse. In particular, strong saturation in the dependence of the amplitude of the established soliton on the amplitude of the initial pulse is found. The results clearly show some similarities and drastic differences between the SSM and the ordinary soliton model based on the nonlinear Schrödinger equation. A random version of the SSM is introduced, with the length of the system’s cell uniformly distributed in some interval, which is a relevant case for applications to fiber-optic telecommunication networks. It is found that the dynamics of the SSM solitons as well as interactions between them in random systems (both single-channel and multichannel systems) are virtually the same as in their regular counterparts.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  9. L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. J. Atai and B. A. Malomed, “Solitary waves in systems with separated Bragg grating and nonlinearity,” Phys. Rev. E 64, 066617 (2001).
    [CrossRef]
  31. 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]

2003 (2)

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

2002 (6)

J. Garnier, “Stabilization of dispersion-managed solitons in random optical fibers by strong dispersion management,” Opt. Commun. 206, 411–438 (2002).
[CrossRef]

R. Driben and B. A. Malomed, “Equilibrium and nonequilibrium solitons in a lossy split-step system with lumped amplification,” Phys. Lett. A 301, 19–26 (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]

A. Kaplan, B. V. Gisin, and B. A. Malomed, “Stable propagation and all-optical switching in planar waveguide-antiwaveguide periodic structures,” J. Opt. Soc. Am. B 19, 522–528 (2002).
[CrossRef]

I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 537–543 (2002).
[CrossRef]

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69–191 (2002).

2001 (5)

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (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]

M. Gutin, U. Mahlab, and B. A. Malomed, “Shaping NRZ pulses and suppression of the inter-symbol interference by a second-harmonic-generating module,” Opt. Commun. 200, 401–414 (2001).
[CrossRef]

R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481–489 (2001).
[CrossRef]

B. A. Malomed and A. Berntson, “Propagation of an optical pulse in a fiber link with random dispersion management,” J. Opt. Soc. Am. B 18, 1243–1251 (2001).
[CrossRef]

2000 (4)

1999 (2)

L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268–1270 (1999).
[CrossRef]

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

1998 (2)

1997 (2)

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

1996 (1)

1995 (1)

M. Nakazawa and H. Kubota, “Optical soliton communication in a positively and negatively dispersion-allocated optical-fiber transmission-line,” Electron. Lett. 31, 216–217 (1995).
[CrossRef]

1994 (1)

1988 (1)

1974 (1)

J. Satsuma and N. Yajima, “Initial value problem for one-dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

F. Kh. Abdullaev and B. B. Baizakov, “Disintegration of a soliton in a dispersion-managed optical communication line with random parameters,” Opt. Lett. 25, 93–95 (2000).
[CrossRef]

Ablowitz, M. J.

Anderson, D.

Atai, J.

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]

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

Baizakov, B. B.

Belanger, P.-A.

Bergé, L.

Berntson, A.

Biondini, G.

Caputo, J. G.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Carrasco, S.

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

Chernikov, S. V.

Christiansen, P. L.

Crasovan, L.-C.

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

Doran, N. J.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Driben, R.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

R. Driben and B. A. Malomed, “Equilibrium and nonequilibrium solitons in a lossy split-step system with lumped amplification,” Phys. Lett. A 301, 19–26 (2002).
[CrossRef]

R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481–489 (2001).
[CrossRef]

R. Driben and B. A. Malomed, “Split-step solitons in long fiber links,” Opt. Commun. 185, 439–456 (2000).
[CrossRef]

Forysiak, W.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Furukawa, Y.

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

Gabitov, I.

Gaididei, Yu. B.

Garnier, J.

J. Garnier, “Stabilization of dispersion-managed solitons in random optical fibers by strong dispersion management,” Opt. Commun. 206, 411–438 (2002).
[CrossRef]

Gisin, B. V.

Gutin, M.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

M. Gutin, U. Mahlab, and B. A. Malomed, “Shaping NRZ pulses and suppression of the inter-symbol interference by a second-harmonic-generating module,” Opt. Commun. 200, 401–414 (2001).
[CrossRef]

Hasegawa, A.

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

Inada, Y.

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

Kaplan, A.

Kashyap, R.

Kinoshita, T.

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

Knox, F. M.

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Kodama, Y.

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

Kraenkel, R. A.

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Kubota, H.

M. Nakazawa and H. Kubota, “Optical soliton communication in a positively and negatively dispersion-allocated optical-fiber transmission-line,” Electron. Lett. 31, 216–217 (1995).
[CrossRef]

Lisak, M.

Mahlab, U.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

M. Gutin, U. Mahlab, and B. A. Malomed, “Shaping NRZ pulses and suppression of the inter-symbol interference by a second-harmonic-generating module,” Opt. Commun. 200, 401–414 (2001).
[CrossRef]

Malomed, B. A.

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[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]

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69–191 (2002).

R. Driben and B. A. Malomed, “Equilibrium and nonequilibrium solitons in a lossy split-step system with lumped amplification,” Phys. Lett. A 301, 19–26 (2002).
[CrossRef]

A. Kaplan, B. V. Gisin, and B. A. Malomed, “Stable propagation and all-optical switching in planar waveguide-antiwaveguide periodic structures,” J. Opt. Soc. Am. B 19, 522–528 (2002).
[CrossRef]

I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 537–543 (2002).
[CrossRef]

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

B. A. Malomed and A. Berntson, “Propagation of an optical pulse in a fiber link with random dispersion management,” J. Opt. Soc. Am. B 18, 1243–1251 (2001).
[CrossRef]

R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481–489 (2001).
[CrossRef]

M. Gutin, U. Mahlab, and B. A. Malomed, “Shaping NRZ pulses and suppression of the inter-symbol interference by a second-harmonic-generating module,” Opt. Commun. 200, 401–414 (2001).
[CrossRef]

R. Driben and B. A. Malomed, “Split-step solitons in long fiber links,” Opt. Commun. 185, 439–456 (2000).
[CrossRef]

Mezentsev, V. K.

Mihalache, D.

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

Nakazawa, M.

M. Nakazawa and H. Kubota, “Optical soliton communication in a positively and negatively dispersion-allocated optical-fiber transmission-line,” Electron. Lett. 31, 216–217 (1995).
[CrossRef]

Nijhof, J. H. B.

A. Berntson, N. J. Doran, W. Forysiak, and J. H. B. Nijhof, “Power dependence of dispersion-managed solitons for anomalous, zero, and normal path-average dispersion,” Opt. Lett. 23, 900–902 (1998).
[CrossRef]

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

Paré, C.

Rasmussen, J. Juul

Reichel, T.

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problem for one-dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Taylor, J. R.

Toda, H.

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

Torner, L.

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268–1270 (1999).
[CrossRef]

Torres, J.-P.

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

Towers, I.

Turitsyn, S. K.

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problem for one-dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Electron. Lett. (2)

J. H. B. Nijhof, N. J. Doran, W. Forysiak, and F. M. Knox, “Stable soliton-like propagation in dispersion managed systems with net anomalous, zero and normal dispersion,” Electron. Lett. 33, 1726–1727 (1997).
[CrossRef]

M. Nakazawa and H. Kubota, “Optical soliton communication in a positively and negatively dispersion-allocated optical-fiber transmission-line,” Electron. Lett. 31, 216–217 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11, 1268–1270 (1999).
[CrossRef]

H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, and A. Hasegawa, “Optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEEE Photon. Technol. Lett. 9, 1415–1417 (1997).
[CrossRef]

IEICE Trans. Commun. (1)

H. Toda, Y. Inada, Y. Kodama, and A. Hasegawa, “10 Gbit/s optical soliton transmission experiment in a comb-like dispersion profiled fiber loop,” IEICE Trans. Commun. E82-B, 1541–1543 (1999).

J. Opt. Soc. Am. B (4)

Opt. Commun. (6)

J. Garnier, “Stabilization of dispersion-managed solitons in random optical fibers by strong dispersion management,” Opt. Commun. 206, 411–438 (2002).
[CrossRef]

R. Driben and B. A. Malomed, “Split-step solitons in long fiber links,” Opt. Commun. 185, 439–456 (2000).
[CrossRef]

M. Gutin, U. Mahlab, and B. A. Malomed, “Shaping NRZ pulses and suppression of the inter-symbol interference by a second-harmonic-generating module,” Opt. Commun. 200, 401–414 (2001).
[CrossRef]

R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, “Implementation of nonlinearity management for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules,” Opt. Commun. 218, 93–104 (2003).
[CrossRef]

R. Driben and B. A. Malomed, “Suppression of crosstalk between solitons in a multi-channel split-step system,” Opt. Commun. 197, 481–489 (2001).
[CrossRef]

L. Torner, S. Carrasco, J.-P. Torres, L.-C. Crasovan, and D. Mihalache, “Tandem light bullets,” Opt. Commun. 199, 277–281 (2001).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (2)

R. Driben and B. A. Malomed, “Equilibrium and nonequilibrium solitons in a lossy split-step system with lumped amplification,” Phys. Lett. A 301, 19–26 (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]

Phys. Rev. A (1)

F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67, 013605 (2003).
[CrossRef]

Phys. Rev. E (1)

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

Prog. Opt. (1)

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69–191 (2002).

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problem for one-dimensional self modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic San Diego, Calif., 1995).

V. V. Konotop and L. Vázquez, Nonlinear Random Waves (World Scientific: Singapore, 1994).

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

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Figures (10)

Fig. 1
Fig. 1

Continuous curve: the largest amplitude Amax of the soliton in the periodic SSM as predicted by the variational approximation [see Eq. (30)] versus amplitude η of the initial pulse in Eq. (13) for LD=LN=1/2. Dashed curve: the share of the initial energy that remains trapped in the established soliton after completion of its relaxation, as found from numerical data (residual energy was used to generate the continuous curve). Stars represent a set of values of the amplitude of the established soliton at midpoints of dispersive segments as found from direct simulations.

Fig. 2
Fig. 2

Relaxation of the initial pulse in Eq. (13) with amplitude η in an established soliton: (a) η=1, (b) η=2, (c) η=3, (d) η=4. In this figure and in Fig. 3, the local power, |u(z, τ)|2, is shown versus τ at each midpoint of the dispersive segment. The onset of saturation in the dependence of the amplitude of the established soliton on the amplitude of the initial pulse is clearly seen.

Fig. 3
Fig. 3

Example of the abrupt destruction of a soliton when amplitude η of the initial pulse in Eq. (13) exceeds the value that limits the stability region of the solitons: (a) η=6.00, (b) η=6.10, (c) η=6.18.

Fig. 4
Fig. 4

Diagram showing different outcomes of the evolution starting with the initial configuration in Eq. (14) in the periodic SSM with LD=LN=1/2. The white area shows where the initial pulse completely decays into radiation. The dashed horizontal line, W=3/2, is the exact breather-generation threshold in the corresponding averaged NLS equation.

Fig. 5
Fig. 5

Typical example of splitting of the initial pulse in Eq. (14) with η=2 and W=3 into a set of stable moving breathers.

Fig. 6
Fig. 6

Example of formation of a stable quasi-periodic breather from the initial pulse in Eq. (14) for η=0.4 and W=3.

Fig. 7
Fig. 7

Splitting of a (long-lived) breather formed from the initial pulse in Eq. (14) for η=0.5 and W=3 (which is close to the breathers’ instability border in Fig. 4). The splitting is shown by means of contour plots of |u(z, τ)|2.

Fig. 8
Fig. 8

(a) Typical example of the formation of a soliton in a random SSM with Lmin=1 and Lmax=5 from the initial pulse in Eq. (13) with η=1. (b) For comparison, the same is shown in a regular system with L=(1/2) (Lmax+Lmin)3.

Fig. 9
Fig. 9

Collisions between two solitons in Eq. (13) with η=0.5 and velocities ±0.1 in a random SSM. In this and in Fig. 10, the evolution is displayed by means of contour plots: (a) two in-phase solitons and (b) solitons with initial phase difference π.

Fig. 10
Fig. 10

Example of stable copropagation of two solitons in a random SSM. The solitons were launched as two pulses in Eq. (13) with η=0.5, the temporal separation between them slightly exceeds the critical value.

Equations (31)

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Lmin<L<Lmax.
ivz+(1/2)vττ=0,
iuz+|u|2u=-iαNu,
u(z, τ)=u(0, τ)exp-αNz+i |u(0, τ)|22αN×[1-exp(-2αNz)].
u(z, τ)=u(0, τ)exp[i|u(0, τ)|2z].
u(τ)u(τ)eG,
G=LNαN+LDαD.
LN(eff)=(2αN)-1([1-exp(-2αNz0)]+exp(2G-2αNz0){1-exp[-2αN(LN-z0)]}),
E12-+|u(0, τ)|2dτ,
P=i-+(uτ*u-uτu*)dτ.
ττ/ΛD,zz/ΛD2,uΛNu,zz/ΛN2
2iuz+12 uττ+|u|2u=0.
u0(τ)=η sech(ητ),
u0(τ)=η sech(ητ/W),
u(z, τ)u(z, τ-cz˜)exp(-c2z˜/2+icτ),
u(z, τ)=A(z)sech[τ/a(z)]exp{i[ϕ(z)+b(z)τ2]},
dbdz=-2π2Ea3,dadz=0,dEdz=0,
E=A2a
d2adz2=4π21a3,b=12adadz,dEdz=0.
a=constamax,b(z)=-(2E/π2amax3)(z-zN),
a(z)=(πamin)-1[(πamin2)2+4(z-zD)2]1/2,
b=2(z-zD)(πamin2)2+4(z-zD)2,
(Δb)N=-(2E/π2amax3)LN
(Δb)D=2LD/[(πamin2)2+LD2].
amax=(πamin)-1[(πamin2)2+LD2]1/2.
π LNLD E2 amin6=π2amin4+LD2.
π LNLD Eamax3=π2amax2+(LNE)2.
|u(τ)|=LD/LNa0-1sech(τ/a0),
E=(LD/LN)a0-1.
amin3(LD2/πLN)E-1,amax3(LNLD/π)E.
LD2(Amax4)3+π2E4Amax4=π2(LN/LD)2E8.

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