Abstract

We performed a detailed investigation of the stability of analytic pulselike solutions of the quintic complex Ginzburg–Landau equation that describes the dynamics of the field in a passively mode-locked laser. We found that in general they are unstable except in a few special cases. We also obtained regions in the parameter space in which stable pulse solutions exist. These stable solutions do not have analytical expressions and must be calculated numerically. We compared and connected the regions in which stable solitonlike solutions exist with the lines for which we had analytical solutions.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65 (1993).
    [Crossref]
  2. E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am. B 6, 1736 (1989).
    [Crossref]
  3. 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 (1994); “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350 (1995).
    [Crossref] [PubMed]
  4. A. Mekozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841 (1991); “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350 (1992).
    [Crossref]
  5. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31 (1992).
    [Crossref] [PubMed]
  6. M. Romagnoli, S. Wabnitz, and M. Midrio, “Bandwidth limits of soliton transmission with sliding filters,” Opt. Commun. 104, 293 (1994).
    [Crossref]
  7. V. V. Afanasjev, “Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification,” Opt. Lett. 18, 790 (1993); “Soliton singularity in the system with nonlinear gain,” Opt. Lett. 20, 704 (1995).
    [Crossref] [PubMed]
  8. D. Atkinson, W. Loh, V. V. Afanasjev, A. B. Grudinin, A. J. Seeds, and D. N. Payne, “Increased amplifier spacing in a soliton system with quantum-well saturable absorbers and spectral filtering,” Opt. Lett. 19, 1514 (1994).
    [Crossref] [PubMed]
  9. M. Matusmoto, H. Ikeda, T. Uda, and A. Hasegawa, “Stable soliton transmission in the system with nonlinear gain,” J. Lightwave Technol. 13, 658 (1995).
    [Crossref]
  10. G. P. Agrawal, “Optical pulse propagation in doped fiber amplifier,” Phys. Rev. A 44, 7495 (1991); “Effect of two-photon absorption on the amplification of ultrashort optical pulses,” Phys. Rev. E 48, 2316 (1993).
    [Crossref]
  11. R. Graham, in Fluctuations, Instabilities, and Phase Transitions, T. Riste, ed. (Springer, New York, 1975).
  12. J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
    [Crossref] [PubMed]
  13. M. J. Landman, “Solutions of the Ginzburg–Landau equation of interest in shear flow transition,” Stud. Appl. Math 76, 187 (1987).
  14. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068 (1991).
    [Crossref]
  15. L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 41, 6187 (1991).
    [Crossref]
  16. J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153 (1994).
    [Crossref]
  17. F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
    [Crossref]
  18. C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
    [Crossref]
  19. W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg–Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64, 749 (1990).
    [Crossref] [PubMed]
  20. W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303 (1992).
    [Crossref]
  21. P. Marcq, H. Chaté, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305 (1994).
    [Crossref]
  22. R. Conte and M. Musette, “Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation,” Physica D 69, 1 (1993).
    [Crossref]
  23. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
    [Crossref]
  24. N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].
  25. S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282 (1990); O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. (Paris) 49, 1829 (1988).
    [Crossref] [PubMed]
  26. V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
    [Crossref]
  27. C. R. Doerr, H. A. Haus, and E. P. Ippen, “Additive-pulse limiting,” Opt. Lett. 19, 31 (1994).
    [Crossref] [PubMed]
  28. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “Sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575 (1992).
    [Crossref] [PubMed]

1995 (2)

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

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
[Crossref]

1994 (6)

1993 (3)

R. Conte and M. Musette, “Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation,” Physica D 69, 1 (1993).
[Crossref]

V. V. Afanasjev, “Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification,” Opt. Lett. 18, 790 (1993); “Soliton singularity in the system with nonlinear gain,” Opt. Lett. 20, 704 (1995).
[Crossref] [PubMed]

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65 (1993).
[Crossref]

1992 (3)

1991 (5)

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

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

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 41, 6187 (1991).
[Crossref]

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifier,” Phys. Rev. A 44, 7495 (1991); “Effect of two-photon absorption on the amplification of ultrashort optical pulses,” Phys. Rev. E 48, 2316 (1993).
[Crossref]

J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
[Crossref] [PubMed]

1990 (3)

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

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282 (1990); O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. (Paris) 49, 1829 (1988).
[Crossref] [PubMed]

V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
[Crossref]

1989 (2)

C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
[Crossref]

E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am. B 6, 1736 (1989).
[Crossref]

1987 (2)

M. J. Landman, “Solutions of the Ginzburg–Landau equation of interest in shear flow transition,” Stud. Appl. Math 76, 187 (1987).

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
[Crossref]

1985 (1)

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].

Afanasjev, V. V.

Agrawal, G. P.

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifier,” Phys. Rev. A 44, 7495 (1991); “Effect of two-photon absorption on the amplification of ultrashort optical pulses,” Phys. Rev. E 48, 2316 (1993).
[Crossref]

Akhmediev, N. N.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
[Crossref]

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].

Atkinson, D.

Bélanger, P. A.

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 41, 6187 (1991).
[Crossref]

C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
[Crossref]

Chaté, H.

P. Marcq, H. Chaté, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305 (1994).
[Crossref]

Chen, C.-J.

Conte, R.

P. Marcq, H. Chaté, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305 (1994).
[Crossref]

R. Conte and M. Musette, “Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation,” Physica D 69, 1 (1993).
[Crossref]

Doerr, C. R.

Eleonskii, V. M.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
[Crossref]

Evangelides, S. G.

Fauve, S.

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282 (1990); O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. (Paris) 49, 1829 (1988).
[Crossref] [PubMed]

Fujimoto, J. G.

Gagnon, L.

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 41, 6187 (1991).
[Crossref]

C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
[Crossref]

Gordon, J. P.

Graham, R.

R. Graham, in Fluctuations, Instabilities, and Phase Transitions, T. Riste, ed. (Springer, New York, 1975).

Grudinin, A. B.

Hakim, V.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
[Crossref]

Hasegawa, A.

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

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

Haus, H. A.

Hohenberg, P. C.

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

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

Ikeda, H.

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

Ippen, E. P.

Jakobsen, P.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
[Crossref]

Jones, C. K. R. T.

J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
[Crossref] [PubMed]

Khatri, F. I.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
[Crossref]

Kodama, Y.

Korneev, V. I.

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].

Kulagin, N. E.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
[Crossref]

Kuz’menko, Yu. V.

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].

Lai, Y.

Landman, M. J.

M. J. Landman, “Solutions of the Ginzburg–Landau equation of interest in shear flow transition,” Stud. Appl. Math 76, 187 (1987).

Lenz, G.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
[Crossref]

Liu, L. Y.

Loh, W.

Marcq, P.

P. Marcq, H. Chaté, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305 (1994).
[Crossref]

Matusmoto, M.

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

Mekozzi, A.

Menyuk, C. R.

Midrio, M.

M. Romagnoli, S. Wabnitz, and M. Midrio, “Bandwidth limits of soliton transmission with sliding filters,” Opt. Commun. 104, 293 (1994).
[Crossref]

Mollenauer, L. F.

Moores, J. D.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
[Crossref]

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153 (1994).
[Crossref]

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65 (1993).
[Crossref]

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

Musette, M.

R. Conte and M. Musette, “Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation,” Physica D 69, 1 (1993).
[Crossref]

Newell, A. C.

J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
[Crossref] [PubMed]

Paré, C.

C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
[Crossref]

Payne, D. N.

Pomeau, Y.

V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
[Crossref]

Powell, J. A.

J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
[Crossref] [PubMed]

Romagnoli, M.

M. Romagnoli, S. Wabnitz, and M. Midrio, “Bandwidth limits of soliton transmission with sliding filters,” Opt. Commun. 104, 293 (1994).
[Crossref]

Seeds, A. J.

Thual, O.

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282 (1990); O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. (Paris) 49, 1829 (1988).
[Crossref] [PubMed]

Uda, T.

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

van Saarloos, W.

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

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

Wabnitz, S.

M. Romagnoli, S. Wabnitz, and M. Midrio, “Bandwidth limits of soliton transmission with sliding filters,” Opt. Commun. 104, 293 (1994).
[Crossref]

Wai, P. K. A.

Wong, W. S.

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153 (1994).
[Crossref]

Europhys. Lett. (1)

V. Hakim, P. Jakobsen, and Y. Pomeau, “Front versus solitary waves in nonequilibrium systems,” Europhys. Lett. 11, 19 (1990).
[Crossref]

J. Lightwave Technol. (1)

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

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

Opt. Commun. (5)

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65 (1993).
[Crossref]

M. Romagnoli, S. Wabnitz, and M. Midrio, “Bandwidth limits of soliton transmission with sliding filters,” Opt. Commun. 104, 293 (1994).
[Crossref]

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153 (1994).
[Crossref]

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447 (1995).
[Crossref]

C. Paré, L. Gagnon, and P. A. Bélanger, “Spatial solitary wave in a weakly saturated amplifying/absorbing medium,” Opt. Commun. 74, 228 (1989).
[Crossref]

Opt. Lett. (7)

V. V. Afanasjev, “Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification,” Opt. Lett. 18, 790 (1993); “Soliton singularity in the system with nonlinear gain,” Opt. Lett. 20, 704 (1995).
[Crossref] [PubMed]

D. Atkinson, W. Loh, V. V. Afanasjev, A. B. Grudinin, A. J. Seeds, and D. N. Payne, “Increased amplifier spacing in a soliton system with quantum-well saturable absorbers and spectral filtering,” Opt. Lett. 19, 1514 (1994).
[Crossref] [PubMed]

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 (1994); “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350 (1995).
[Crossref] [PubMed]

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

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

C. R. Doerr, H. A. Haus, and E. P. Ippen, “Additive-pulse limiting,” Opt. Lett. 19, 31 (1994).
[Crossref] [PubMed]

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

Phys. Rev. A (3)

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 41, 6187 (1991).
[Crossref]

J. A. Powell, A. C. Newell, and C. K. R. T. Jones, “Competition between generic and nongeneric fronts in envelope equations,” Phys. Rev. A 44, 3636 (1991).
[Crossref] [PubMed]

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifier,” Phys. Rev. A 44, 7495 (1991); “Effect of two-photon absorption on the amplification of ultrashort optical pulses,” Phys. Rev. E 48, 2316 (1993).
[Crossref]

Phys. Rev. Lett. (2)

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

S. Fauve and O. Thual, “Solitary waves generated by subcritical instabilities in dissipative systems,” Phys. Rev. Lett. 64, 282 (1990); O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. (Paris) 49, 1829 (1988).
[Crossref] [PubMed]

Physica D (3)

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

P. Marcq, H. Chaté, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg–Landau equation,” Physica D 73, 305 (1994).
[Crossref]

R. Conte and M. Musette, “Linearity inside nonlinearity: exact solutions to the complex Ginzburg–Landau equation,” Physica D 69, 1 (1993).
[Crossref]

Stud. Appl. Math (1)

M. J. Landman, “Solutions of the Ginzburg–Landau equation of interest in shear flow transition,” Stud. Appl. Math 76, 187 (1987).

Theor. Math. Phys. (USSR) (1)

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. (USSR) 72, 809 (1987).
[Crossref]

Zh. Eksp. Teor. Fiz (1)

N. N. Akhmediev, V. I. Korneev, and Yu. V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Zh. Eksp. Teor. Fiz 88, 107 (1985) [Sov. Phys. JETP 61, 62 (1985)].

Other (1)

R. Graham, in Fluctuations, Instabilities, and Phase Transitions, T. Riste, ed. (Springer, New York, 1975).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Curves delimiting the regions on the (β, ) plane for which the stationary solutions [Eq. (21)] studied in this paper exist. The dotted–dashed and the dotted curves are obtained when d = d and ν = −0.5 and when δ = −0.1 and δ = −0.5, respectively. The region in which the solution exists is the area above the curves. The dashed curve is for d = d+ and for negative values of δ and ν. The allowed region is in this last case the area below the curve. The solid curve represents the curve given by Eq. (16).

Fig. 2
Fig. 2

Dependence of μ on [Eq. (9a)] that must be satisfied as a necessary condition for the general solution to exist when β = 0.5 and ν = −0.5. The solid curve is for d = d+ in Eq. (10), and the dotted curve is for d = d. The horizontal lines mark the intervals at which the solution given by Eqs. (2)(17) exists for a given value of δ, which is written on the lines.

Fig. 3
Fig. 3

Field amplitude of the exact solution (dashed curve) for β = 0.5, ν = −0.5, = 0.6, and δ = −0.1; the real (solid curve) and the imaginary (dotted curve) parts of the perturbation eigenmode associated with this solution.

Fig. 4
Fig. 4

Two possible scenarios of evolution of the stationary pulse represented in Fig. 3.

Fig. 5
Fig. 5

Growth rate of the predominant perturbation eigenmode associated with the soliton solution as a function of for β = 0.5, ν = −0.5, (a) δ < 0, and (b) δ > 0: In both cases the solid curve is for |δ| = 0.001 and d = d, the dotted–dashed curve is for |δ| = 0.1 and d = d, the dotted curve is for |δ| = 0.001 and d = d+, and the dashed curve is for |δ| = 0.1 and d = d+. For clarity, the value of δ is on the corresponding curves.

Fig. 6
Fig. 6

Evolution of the solution for β = 0.5, = 0.1, ν = −0.5, δ = +0.005, and μ = −0.09. This stationary solution is initially perturbed as indicated by Eqs. (24) and (25).

Fig. 7
Fig. 7

Evolution of the solution for β = 0.5, = 0.4006, ν = −0.5, δ = −0.1, and μ = −0.227. This stationary solution is initially perturbed as indicated by Eqs. (24) and (25).

Fig. 8
Fig. 8

Growth rate of the predominant perturbation eigenmode associated with the soliton solution as a function of for ν = −0.1, β = 0.5, β = 0.1, (a) δ = −0.01, and (b) δ = −0.05. The different curves correspond to (i) β = 0.5 and d = d (solid curves), (ii) β = 0.1 and d = d (dotted–dashed curves), (iii) β = 0.5 and d = d+ (dashed curves), (iv) β = 0.1 and d = d+ (dotted curves). For clarity we wrote the values of β in parentheses near the corresponding curves.

Fig. 9
Fig. 9

(a) Dependence of μ on [Eq. (9a)] that must be satisfied as a necessary condition for the general solution Eq. (21) to exist when ν = −0.5 and β = 0.5 (upper curve) and β = 0.1 (lower curve). The solid and the dashed parts of these curves show the regions in which the solutions actually exists for δ = −0.01. (b) The growth rate of the predominant perturbation eigenmode associated with the soliton solution as a function of for δ = −0.01 and (β, ν) equal to (i) (0.5, 0.1) (solid curve), (ii) (0.5, 0.5) (dotted curve), (iii) (0.1, 0.1) (long-dashed curve), and (iv) (0.1, 0.5) (short-dashed curve).

Fig. 10
Fig. 10

Numerically found soliton solutions for β = 0.5, ν = μ = δ = −0.1, = 0.38 (solid curves), = 0.52 (dotted curves) and = 0.66 (dashed curves). (a) Amplitude profile |μ|, (b) phase profile arg (ψ). The circle, the diamond, and the triangle symbols associated with the cases equal to 0.38, 0.52, and 0.66, respectively, are used in the following figures to locate these solutions in the parameter space.

Fig. 11
Fig. 11

Region in the (β) plane in which stable pulselike solutions are found. Differently hatched areas are for different values of μ, which are written in the corresponding regions. All these areas are located above the curves S (dashed curves). (a) δ = −0.01, ν = −0.1, and (b) δ = ν = −0.1.

Fig. 12
Fig. 12

Region in the plane (ν, ) for which stable pulses are possible. β = 0.5 and μ = δ = −0.1. The dashed curve represents the points at which the analytical solution given by Eq. (17) exists. The three symbols (circle, diamond, and triangle) show the locations of the solutions represented in Fig. 10.

Fig. 13
Fig. 13

Region in the plane (δ, ) in which stable pulses are possible. β = 0.5 and ν = μ = −0.1. For these parameters the analytical solution exists at = 1. The three symbols (circle, diamond, and triangle) show the locations of the solutions represented in Fig. 10.

Fig. 14
Fig. 14

Region in the plane (μ, ) in which stable pulses are found. β = 0.5 and ν = δ = −0.1. The dashed line represents the points at which the analytical solution exists.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

i ψ z + 1 2 ψ t t + | ψ | 2 ψ = i δ ψ + i | ψ | 2 ψ + i β ψ t t + i μ | ψ | 4 ψ ν | ψ | 4 ψ ,
ψ ( t , z ) = A ( t ) exp ( i ω z ) ,
A ( t ) = a ( t ) exp [ i ϕ ( t ) ] ,
( ω 1 2 ϕ 2 + β ϕ ) a + 2 β ϕ a + 1 2 a + a 3 + ν a 5 = 0 , ( δ + β ϕ 2 + 1 2 ϕ ) a + ϕ a β a a 3 μ a 5 = 0 ,
ϕ ( t ) = ϕ 0 + d ln [ a ( t ) ] ,
ω a + ( 1 2 + β d ) a + ( β d d 2 2 ) a 2 a + a 3 + ν a 5 = 0 , δ a + ( d 2 β ) a + ( d 2 + β d 2 ) a 2 a a 3 μ a 5 = 0 .
d 4 ( 1 + d 2 ) ( 1 + 4 β 2 ) a 2 a 2 + 1 2 ( d 2 + β d 2 + β d d 2 2 ) a 2 + 1 3 [ ν ( d 2 + β d 2 ) + μ ( β d d 2 2 ) ] a 4 + ω d 2 ( 1 + 2 β d ) + δ ( β d d 2 2 ) = 0 .
d 4 ( 1 + d 2 ) ( 1 + 4 β 2 ) a 2 a 2 + ( β d 2 2 β d ) a 2 [ ν ( d 2 β ) + μ ( β d + 1 2 ) ] a 4 + ω β ω d 2 δ 2 δ β d = 0 .
ν ( 4 d + 2 β d 2 6 β ) + μ ( 8 β d d 2 + 3 ) = 0 ,
3 d + 2 β d 2 4 β + 6 β d + 2 d 2 = 0 ,
2 ω ( d β + β d 2 ) + δ ( 1 d 2 + 4 β d ) = 0 .
d = d ± = 3 ( 1 + 2 β ) ± 9 ( 1 + 2 β ) 2 + 8 ( 2 β ) 2 2 ( 2 β ) .
ω = δ ( 1 d 2 + 4 β d ) 2 ( d β + β d 2 ) .
a 2 a 2 + 2 ν 8 β d d 2 + 3 a 4 + 2 ( 2 β ) 3 d ( 1 + 4 β 2 ) a 2 δ d β + β d 2 = 0 .
2 ν 8 β d d 2 + 3 = μ 3 β 2 d β d 2 .
f 2 f 2 + 8 ν 8 β d d 2 + 3 f 2 + 8 ( 2 β ) 3 d ( 1 + 4 β 2 ) f 4 δ d β + β d 2 = 0 .
4 δ d β + β d 2 > 0 .
S = β 3 1 + 4 β 2 1 4 + 18 β 2 ,
f ( t ) = 2 f 1 f 2 ( f 1 + f 2 ) ( f 1 f 2 ) cosh ( 2 α f 1 | f 2 | t ) ,
α = | 2 ν 8 β d d 2 + 3 | = | μ 3 β 2 d β d 2 | ,
2 ν 8 β d d 2 + 3 f 2 + 2 ( 2 β ) 3 d ( 1 + 4 β 2 ) f δ d β + β d 2 = 0 ,
f 1,2 = ( 2 β ) ± ( 2 β ) 2 + 18 δ d 2 ν ( 1 + 4 β 2 ) 2 ( 8 β d d 2 + 3 ) ( d β + β d 2 ) 6 d ν ( 1 + 4 β 2 ) × ( 8 β d d 2 + 3 ) ,
f 1,2 = ( 2 β ) ± ( 2 β ) 2 + 9 δ d 2 μ ( 1 + 4 β 2 ) 2 ( 3 β 2 d β d 2 ) ( d β + β d 2 ) 3 d μ ( 1 + 4 β 2 ) × ( 3 β 2 d + β d 2 )
ψ ( t , z ) = [ A 0 ( t ) + γ g ( t , z ) ] exp ( i ω z ) ,
i g z + ω g + ( 1 2 i β ) g t t + 2 | A 0 | 2 ( 1 i ) g + A 0 2 ( 1 i ) g * + ( ν i μ ) ( 3 | A 0 | 4 g + 2 | A 0 | 2 A 0 2 g * ) = 0 .
ψ ( t , 0 ) = A 0 ( t ) [ 1 + Γ ( t ) ] ,
< Γ ( t ) > = 0 , < | Γ ( t ) | 2 > = 0.2 .

Metrics