Abstract

Optical-fiber transmission of pulses can be modeled with the complex Ginzburg–Landau equation. We find novel stable soliton pairs and trains, which are relevant in this case, and analyze them. We suggest that the distance between the pulses and the phase difference between them is defined by energy and momentum balance equations, rather than by equations of standard perturbation theory. We present a two-dimensional phase plane (interaction plane) for analyzing the stability properties and general dynamics of two-soliton solutions of the Complex Ginzburg–Landau equation.

© 1998 Optical Society of America

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  1. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991);“Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
    [CrossRef] [PubMed]
  2. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect,” Opt. Lett. 17, 31–34 (1992).
    [CrossRef] [PubMed]
  3. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  4. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  5. M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
    [CrossRef]
  6. P. A. Bélanger, L. Gagnon, and C. Paré, “Solitary pulses in an amplified nonlinear dispersive medium,” Opt. Lett. 14, 943–945 (1989);C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228–232 (1989).
    [CrossRef] [PubMed]
  7. P. A. Bélanger, “Coupled-cavity mode locking: a nonlinear model,” J. Opt. Soc. Am. B 8, 2077–2081 (1991).
    [CrossRef]
  8. J. D. Moores, “On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65–70 (1993).
    [CrossRef]
  9. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
    [CrossRef]
  10. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
    [CrossRef]
  11. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive mode-locking in fiber ring laser: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
    [CrossRef]
  12. M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulator,” Opt. Lett. 16, 502–504 (1991).
    [CrossRef] [PubMed]
  13. V. J. Matsas, D. J. Richardson, T. P. Newson, and D. N. Payne, “Characterization of a self-starting passively modelocked fiber ring laser that exploits nonlinear polarization evolution,” Opt. Lett. 18, 358–360 (1993).
    [CrossRef] [PubMed]
  14. C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Stability of passively mode-locked fiber lasers with fast saturable absorption,” Opt. Lett. 19, 198–200 (1994).
    [CrossRef] [PubMed]
  15. M. Romagnoli, S. Wabnitz, P. Franco, M. Midrio, L. Bossalini, and F. Fontana, “Role of dispersion in pulse emission from a sliding-frequency fiber laser,” J. Opt. Soc. Am. B 12, 938–944 (1995).
    [CrossRef]
  16. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  17. D. Anderson and M. Lisak, “Bandwidth limits due to mutual pulse interaction in optical soliton communication systems,” Opt. Lett. 11, 174–176 (1986).
    [CrossRef] [PubMed]
  18. Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1982 (1992).
    [CrossRef]
  19. V. V. Afanasjev, “Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification,” Opt. Lett. 18, 790–792 (1993).
    [CrossRef] [PubMed]
  20. C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” IEE Proc. J 134, 145–151 (1987).
  21. N. N. Akhmediev and A. Ankiewicz, “Generation of a train of solitons with arbitrary phase difference between neighboring solitons,” Opt. Lett. 19, 545–547 (1994).
    [CrossRef] [PubMed]
  22. N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  23. A. B. Buryak and N. N. Akhmediev, “Stability criterion for stationary bound states of solitons with radiationless oscillating tails,” Phys. Rev. E 51, 3572–3578 (1995).
    [CrossRef]
  24. H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801–2804 (1989).
    [CrossRef] [PubMed]
  25. J. Alexander and C. K. R. T. Jones, “Existence and stability of asymptotically oscillatory double pulses,” J. reine angew. Math. 44649–79 (1994).
  26. B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
    [CrossRef] [PubMed]
  27. The theory of soliton bound states in Ref. 26 was simply (and to some extent quite arbitrarily) based on the introduction of an effective potential of interaction between the solitary pulses related to G. L. Lyapunov’s function.
  28. V. I. Karpman and V. V. Solov’ev, “A perturbation approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
    [CrossRef]
  29. K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
    [CrossRef]
  30. V. V. Afanasjev and N. Akhmediev, “Soliton interaction in nonequilibrium dynamical systems,” Phys. Rev. E 53, 6471–6475 (1996).
    [CrossRef]
  31. N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
    [CrossRef]
  32. W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equation,” Physica D 56, 303–367 (1992).
    [CrossRef]
  33. W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 64, 749–752 (1990).
    [CrossRef] [PubMed]
  34. V. V. Afanasjev, N. N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localised solution of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
    [CrossRef]
  35. N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation,” Phys. Lett. A 110, 133–135 (1985).
    [CrossRef]
  36. V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs. solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19–24 (1990).
    [CrossRef]
  37. J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997).
    [CrossRef]
  38. W. Schöpf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 66, 2316–2319 (1991).
    [CrossRef]
  39. V. V. Afanasjev and N. N. Akhmediev, “A new kind of periodic stationary solution of the cubic Ginzburg-Landau equation,” Physica A 233, 801–808 (1996).
    [CrossRef]
  40. N. N. Akhmediev, G. Town, and S. Wabnitz, “Soliton coding based on shape invariant interacting soliton packets: the three-soliton case,” Opt. Commun. 104, 385–390 (1994).
    [CrossRef]
  41. K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg-Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
    [CrossRef]

1997 (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997).
[CrossRef]

1996 (4)

V. V. Afanasjev and N. N. Akhmediev, “A new kind of periodic stationary solution of the cubic Ginzburg-Landau equation,” Physica A 233, 801–808 (1996).
[CrossRef]

V. V. Afanasjev and N. Akhmediev, “Soliton interaction in nonequilibrium dynamical systems,” Phys. Rev. E 53, 6471–6475 (1996).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

V. V. Afanasjev, N. N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localised solution of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

1995 (4)

A. B. Buryak and N. N. Akhmediev, “Stability criterion for stationary bound states of solitons with radiationless oscillating tails,” Phys. Rev. E 51, 3572–3578 (1995).
[CrossRef]

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive mode-locking in fiber ring laser: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

M. Romagnoli, S. Wabnitz, P. Franco, M. Midrio, L. Bossalini, and F. Fontana, “Role of dispersion in pulse emission from a sliding-frequency fiber laser,” J. Opt. Soc. Am. B 12, 938–944 (1995).
[CrossRef]

1994 (5)

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Stability of passively mode-locked fiber lasers with fast saturable absorption,” Opt. Lett. 19, 198–200 (1994).
[CrossRef] [PubMed]

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, “Generation of a train of solitons with arbitrary phase difference between neighboring solitons,” Opt. Lett. 19, 545–547 (1994).
[CrossRef] [PubMed]

J. Alexander and C. K. R. T. Jones, “Existence and stability of asymptotically oscillatory double pulses,” J. reine angew. Math. 44649–79 (1994).

N. N. Akhmediev, G. Town, and S. Wabnitz, “Soliton coding based on shape invariant interacting soliton packets: the three-soliton case,” Opt. Commun. 104, 385–390 (1994).
[CrossRef]

1993 (3)

1992 (4)

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect,” Opt. Lett. 17, 31–34 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
[CrossRef] [PubMed]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

1991 (6)

1990 (2)

V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs. solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19–24 (1990).
[CrossRef]

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

1989 (2)

1987 (1)

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” IEE Proc. J 134, 145–151 (1987).

1986 (1)

1985 (1)

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation,” Phys. Lett. A 110, 133–135 (1985).
[CrossRef]

1984 (1)

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg-Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

1983 (1)

1981 (2)

V. I. Karpman and V. V. Solov’ev, “A perturbation approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Afanasjev, V. V.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997).
[CrossRef]

V. V. Afanasjev and N. N. Akhmediev, “A new kind of periodic stationary solution of the cubic Ginzburg-Landau equation,” Physica A 233, 801–808 (1996).
[CrossRef]

V. V. Afanasjev and N. Akhmediev, “Soliton interaction in nonequilibrium dynamical systems,” Phys. Rev. E 53, 6471–6475 (1996).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

V. V. Afanasjev, N. N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localised solution of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

V. V. Afanasjev, “Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification,” Opt. Lett. 18, 790–792 (1993).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

Akhmediev, N.

V. V. Afanasjev and N. Akhmediev, “Soliton interaction in nonequilibrium dynamical systems,” Phys. Rev. E 53, 6471–6475 (1996).
[CrossRef]

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997).
[CrossRef]

V. V. Afanasjev and N. N. Akhmediev, “A new kind of periodic stationary solution of the cubic Ginzburg-Landau equation,” Physica A 233, 801–808 (1996).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

V. V. Afanasjev, N. N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localised solution of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

A. B. Buryak and N. N. Akhmediev, “Stability criterion for stationary bound states of solitons with radiationless oscillating tails,” Phys. Rev. E 51, 3572–3578 (1995).
[CrossRef]

N. N. Akhmediev, G. Town, and S. Wabnitz, “Soliton coding based on shape invariant interacting soliton packets: the three-soliton case,” Opt. Commun. 104, 385–390 (1994).
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, “Generation of a train of solitons with arbitrary phase difference between neighboring solitons,” Opt. Lett. 19, 545–547 (1994).
[CrossRef] [PubMed]

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

Alexander, J.

J. Alexander and C. K. R. T. Jones, “Existence and stability of asymptotically oscillatory double pulses,” J. reine angew. Math. 44649–79 (1994).

Anderson, D.

Ankiewicz, A.

Bekki, N.

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation,” Phys. Lett. A 110, 133–135 (1985).
[CrossRef]

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg-Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Bélanger, P. A.

Bossalini, L.

Brand, H. R.

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801–2804 (1989).
[CrossRef] [PubMed]

Buryak, A. B.

A. B. Buryak and N. N. Akhmediev, “Stability criterion for stationary bound states of solitons with radiationless oscillating tails,” Phys. Rev. E 51, 3572–3578 (1995).
[CrossRef]

Chen, C.-J.

Chu, P. L.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” IEE Proc. J 134, 145–151 (1987).

Deissler, R. J.

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801–2804 (1989).
[CrossRef] [PubMed]

Desem, C.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” IEE Proc. J 134, 145–151 (1987).

Evangelides, S. G.

Fermann, M. E.

Fontana, F.

Franco, P.

Fujimoto, J. G.

Gagnon, L.

Gordon, J. P.

Gorshkov, K. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Haberl, F.

Hakim, V.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs. solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19–24 (1990).
[CrossRef]

Hasegawa, A.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect,” Opt. Lett. 17, 31–34 (1992).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive mode-locking in fiber ring laser: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991);“Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
[CrossRef] [PubMed]

Hofer, M.

Hohenberg, P. C.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equation,” Physica D 56, 303–367 (1992).
[CrossRef]

W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation,” Phys. Rev. Lett. 64, 749–752 (1990).
[CrossRef] [PubMed]

Ikeda, H.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Ippen, E. P.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive mode-locking in fiber ring laser: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Jakobsen, P.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs. solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19–24 (1990).
[CrossRef]

Jones, C. K. R. T.

J. Alexander and C. K. R. T. Jones, “Existence and stability of asymptotically oscillatory double pulses,” J. reine angew. Math. 44649–79 (1994).

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbation approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect,” Opt. Lett. 17, 31–34 (1992).
[CrossRef] [PubMed]

Kramer, L.

W. Schöpf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 66, 2316–2319 (1991).
[CrossRef]

Lai, Y.

Lisak, M.

Malomed, B. A.

B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954–6957 (1991).
[CrossRef] [PubMed]

Matsas, V. J.

Matsumoto, M.

M. Matsumoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658–665 (1995).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Midrio, M.

Mollenauer, L. F.

Moores, J. D.

Nelson, L. E.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive mode-locking in fiber ring laser: theory and experiment,” IEEE J. Quantum Electron. 31, 591–598 (1995).
[CrossRef]

Newson, T. P.

Nozaki, K.

N. Bekki and K. Nozaki, “Formations of spatial patterns and holes in the generalized Ginzburg-Landau equation,” Phys. Lett. A 110, 133–135 (1985).
[CrossRef]

K. Nozaki and N. Bekki, “Exact solutions of the generalized Ginzburg-Landau equation,” J. Phys. Soc. Jpn. 53, 1581–1582 (1984).
[CrossRef]

Ober, M. H.

Ostrovsky, L. A.

K. A. Gorshkov and L. A. Ostrovsky, “Interactions of solitons in nonintegrable systems: direct perturbation method and applications,” Physica D 3, 428–438 (1981).
[CrossRef]

Paré, C.

Payne, D. N.

Pomeau, Y.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs. solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19–24 (1990).
[CrossRef]

Richardson, D. J.

Romagnoli, M.

Schmidt, A. J.

Schöpf, W.

W. Schöpf and L. Kramer, “Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 66, 2316–2319 (1991).
[CrossRef]

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbation approach to the two-soliton systems,” Physica D 3, 487–502 (1981).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783–4796 (1997).
[CrossRef]

V. V. Afanasjev, N. N. Akhmediev, and J. M. Soto-Crespo, “Three forms of localised solution of the quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1931–1939 (1996).
[CrossRef]

N. N. Akhmediev, V. V. Afanasjev, and J. M. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190–1201 (1996).
[CrossRef]

Tamura, K.

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Other (3)

The theory of soliton bound states in Ref. 26 was simply (and to some extent quite arbitrarily) based on the introduction of an effective potential of interaction between the solitary pulses related to G. L. Lyapunov’s function.

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

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Figures (13)

Fig. 1
Fig. 1

Functionals (a) F[ψ0, ρ, ϕ] and (b) J[ψ0, ρ, ϕ] in terms of the separation ρ for ϕ fixed at π/2. The parameters of the simulation are shown in the figure. The inset in (b) shows the functional J on a different scale to reveal an additional zero of J[ψ0, ρ, ϕ].

Fig. 2
Fig. 2

Zeros of F[ψ0, ρ, ϕ] (solid curves) and J[ψ0, ρ, ϕ] (dotted curves) on the interaction plane. The points of intersection of the solid curves with the dotted curves correspond to the bound states of two solitons. They are shown as filled circles. The parameters of the simulation are shown in the figure.

Fig. 3
Fig. 3

Trajectories showing the evolution of two-soliton solutions on the interaction plane. The five singular points correspond to the five bound states depicted in Fig. 2. Only two of them (F1 and F2) are stable. The central part of the figure, where ρ is less than a single soliton width, does not describe a valid bound state. Trajectories converging to the center describe the merging of two solitons. The parameters used in the simulation are shown in the figure.

Fig. 4
Fig. 4

(a) Stable propagation of a two-soliton solution and (b) its spectrum. The parameters of the simulation are the same as in Fig. 3.

Fig. 5
Fig. 5

Trajectories showing the evolution of two-soliton solutions on the interaction plane. The two singular points (F1 and F2) are now (a) neutrally stable and (b) unstable. The parameters used in the simulation are shown in the figure.

Fig. 6
Fig. 6

(a) Amplitude profile and (b) phase profile of the two-soliton solution. The initial condition, consisting of two solitons at a separation ρ=2.4 and with a ϕ=π/2 phase difference between them, is also given here. The two curves are indistinguishable on the scale of this plot. Note that the phases of the peaks in (b) differ by π/2.

Fig. 7
Fig. 7

Oscillations of the functionals (a) J and (b) F with (c) the energy in the process of convergence of the initial condition (8) into a bound state of two solitons. The parameters used are given in the plot for J.

Fig. 8
Fig. 8

Comparison of the profiles of the two-soliton bound state of the quintic CGLE (solid curves) and the dark-soliton solution of the cubic equation (dotted curves). The inset shows the dark soliton in a much wider time window.

Fig. 9
Fig. 9

Evolution of a three-soliton bound state.

Fig. 10
Fig. 10

Stable propagation of a four-soliton bound state.

Fig. 11
Fig. 11

Creation of a soliton train from a single pulse. The parameters of the simulation are shown on the plot.

Fig. 12
Fig. 12

(a) Amplitude and (b) phase profiles of a three-soliton bound state with zero velocity. Each curve is labeled with its value of .

Fig. 13
Fig. 13

Parameters in the solution (A1) versus κ. Note that w=V=K=0 when κ=0.02579. This point corresponds to Eq. (A9).

Equations (21)

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iψξ+D2 ψττ+|ψ|2ψ=iδψ+i|ψ|2ψ+iβψττ+iμ|ψ|4ψ-ν|ψ|4ψ,
ddξ -|ψ|2dτ=F[ψ],
F[ψ]=2-[δ|ψ|2+|ψ|4+μ|ψ|6-β|ψτ|2]dτ.
i2ddξ -(ψτψ*-ψτ*ψ)dτ=J[ψ],
J[ψ]=i-[(δ+|ψ|2+μ|ψ|4)(ψτψ*-ψτ*ψ)+β(ψτψττ*-ψτ*ψττ)]dτ.
F[ψ]=0,
J[ψ]=0.
ψ(τ)=ψ0(τ-ρ/2)+ψ0(τ+ρ/2)exp(iϕ),
F[ψ0, ρ, ϕ]=0,
J[ψ0, ρ, ϕ]=0.
ψ(τ, ξ)=κu tanh(κζ)+(d+2i) wd×exp[iϕ(ζ)] exp(iKζ-iΩξ),
ϕ(ζ)=d ln[cosh(κζ)].
d=3(2β+1)+9(2β+1)2+8(2β-)22(2β-).
u2=3d(1+4β2)2(2β-)=(1+4β2)(d2-2)2(1+2β).
wd2=β κ2(3d+4β)-2δ3(d+4β),
Ω=δ2β+2β-2 K2-κ2d2+δβ,
K=uβ wd,
K2=d2β (1+4β2)(2β-) [κ2(3d+4β)-2δ](d+4β),
V=K (-2β).
κ2=2δ3d+4β,
Ω=κ2(1-3βd)=2δ(1-3βd)3d+4β.

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