Abstract

The N-soliton solutions for the integrable Hirota equation describing pulse propagation in optical fibers with higher-order effects are presented by using the Darboux transformation method. As an example, the general one-soliton solution on a cw background is given in its explicit form. Then, two exact analytic solutions that describe (i) modulation instability and (ii) bright pulse propagation on a cw background are discussed in detail. The simulations performed in selected cases show that these soliton solutions can be generated numerically when the involved parameters do not exactly satisfy the required integrability conditions.

© 2004 Optical Society of America

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    [CrossRef]
  2. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, UK, 1995).
  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
  4. Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
    [CrossRef]
  5. R. Hirota, “Exact envelope-soliton solution of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
    [CrossRef]
  6. N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
    [CrossRef]
  7. D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
    [CrossRef]
  8. D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
    [CrossRef]
  9. K. Porsezian and K. Nakkeeran, “Optical solitons in presence of Kerr dispersion and self-frequency shift,” Phys. Rev. Lett. 76, 3955–3958 (1996).
    [CrossRef] [PubMed]
  10. M. Gedalin, T. C. Scott, and Y. B. Band, “Optical solitary waves in the higher-order nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 448–451 (1997).
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  11. D. Mihalache, N. Truta, and L. C. Crasovan, “Painleve analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term,” Phys. Rev. E 56, 1064–1070 (1997).
    [CrossRef]
  12. Z. H. Li, G. S. Zhou, and D. C. Su, “N-soliton solution in the higher order nonlinear Schrödinger equation,” in Fiber Optic Components and Optical Communication II, S. Jian, F. F. Tong, and R. Maerz, eds., Proc. SPIE 3552, 226–231 (1998).
    [CrossRef]
  13. Y. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
    [CrossRef] [PubMed]
  14. R. Radhakrishnam and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
    [CrossRef]
  15. S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
    [CrossRef]
  16. A. Mahalingam and K. Porsezian, “Propagation of dark solitons with higher-order effects in optical fibers,” Phys. Rev. E 64, 046608 (2001).
    [CrossRef]
  17. L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
    [CrossRef]
  18. Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
    [CrossRef] [PubMed]
  19. W. P. Hong, “Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms,” Opt. Commun. 194, 217–223 (2001).
    [CrossRef]
  20. N. N. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
    [CrossRef]
  21. Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
    [CrossRef]
  22. L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
    [CrossRef]
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  24. Q. H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. 82, 4432–4435 (1999).
    [CrossRef]
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  26. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef] [PubMed]
  27. D. Mihalache and N. C. Panoiu, “Analytic method for solving the nonlinear Schrödinger equation describing pulse propagation in dispersive optic fibres,” J. Phys. A: Math. Gen. 26, 2679–2697 (1993).
    [CrossRef]
  28. D. Mihalache, F. Lederer, and D.-M. Baboiu, “Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation,” Phys. Rev. A 47, 3285–3290 (1993).
    [CrossRef] [PubMed]
  29. L. Gagnon, “Solitons on a continuous-wave background and collision between two dark pulses: some analytical results,” J. Opt. Soc. Am. B 10, 469–474 (1993).
    [CrossRef]

2004

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

2003

Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[CrossRef]

2002

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

2001

A. Mahalingam and K. Porsezian, “Propagation of dark solitons with higher-order effects in optical fibers,” Phys. Rev. E 64, 046608 (2001).
[CrossRef]

W. P. Hong, “Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms,” Opt. Commun. 194, 217–223 (2001).
[CrossRef]

2000

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

1999

S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Q. H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. 82, 4432–4435 (1999).
[CrossRef]

1998

Z. H. Li, G. S. Zhou, and D. C. Su, “N-soliton solution in the higher order nonlinear Schrödinger equation,” in Fiber Optic Components and Optical Communication II, S. Jian, F. F. Tong, and R. Maerz, eds., Proc. SPIE 3552, 226–231 (1998).
[CrossRef]

1997

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical solitary waves in the higher-order nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 448–451 (1997).
[CrossRef]

D. Mihalache, N. Truta, and L. C. Crasovan, “Painleve analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term,” Phys. Rev. E 56, 1064–1070 (1997).
[CrossRef]

1996

K. Porsezian and K. Nakkeeran, “Optical solitons in presence of Kerr dispersion and self-frequency shift,” Phys. Rev. Lett. 76, 3955–3958 (1996).
[CrossRef] [PubMed]

R. Radhakrishnam and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

1994

D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
[CrossRef]

1993

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

D. Mihalache and N. C. Panoiu, “Analytic method for solving the nonlinear Schrödinger equation describing pulse propagation in dispersive optic fibres,” J. Phys. A: Math. Gen. 26, 2679–2697 (1993).
[CrossRef]

D. Mihalache, F. Lederer, and D.-M. Baboiu, “Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation,” Phys. Rev. A 47, 3285–3290 (1993).
[CrossRef] [PubMed]

L. Gagnon, “Solitons on a continuous-wave background and collision between two dark pulses: some analytical results,” J. Opt. Soc. Am. B 10, 469–474 (1993).
[CrossRef]

1991

N. N. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Y. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

1987

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

1973

R. Hirota, “Exact envelope-soliton solution of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Afanasjev, V. V.

Y. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

Akhmediev, N. N.

N. N. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

Baboiu, D.-M.

D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
[CrossRef]

D. Mihalache, F. Lederer, and D.-M. Baboiu, “Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation,” Phys. Rev. A 47, 3285–3290 (1993).
[CrossRef] [PubMed]

Band, Y. B.

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical solitary waves in the higher-order nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 448–451 (1997).
[CrossRef]

Crasovan, L. C.

D. Mihalache, N. Truta, and L. C. Crasovan, “Painleve analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term,” Phys. Rev. E 56, 1064–1070 (1997).
[CrossRef]

Crespo, R. D.

S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Fernandez-Diaz, J. M.

S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Gagnon, L.

Gedalin, M.

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical solitary waves in the higher-order nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 448–451 (1997).
[CrossRef]

Guinea, A.

S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Hirota, R.

R. Hirota, “Exact envelope-soliton solution of a nonlinear wave equation,” J. Math. Phys. 14, 805–809 (1973).
[CrossRef]

Hong, W. P.

W. P. Hong, “Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms,” Opt. Commun. 194, 217–223 (2001).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and V. V. Afanasjev, “Dark optical solitons with reverse-sign amplitude,” Phys. Rev. A 44, R1446–R1449 (1991).
[CrossRef] [PubMed]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

Lakshmanan, M.

R. Radhakrishnam and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

Lederer, F.

D. Mihalache, F. Lederer, and D.-M. Baboiu, “Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation,” Phys. Rev. A 47, 3285–3290 (1993).
[CrossRef] [PubMed]

Li, L.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[CrossRef]

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Li, S. Q.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Li, Z. H.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[CrossRef]

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Z. H. Li, G. S. Zhou, and D. C. Su, “N-soliton solution in the higher order nonlinear Schrödinger equation,” in Fiber Optic Components and Optical Communication II, S. Jian, F. F. Tong, and R. Maerz, eds., Proc. SPIE 3552, 226–231 (1998).
[CrossRef]

Mahalingam, A.

A. Mahalingam and K. Porsezian, “Propagation of dark solitons with higher-order effects in optical fibers,” Phys. Rev. E 64, 046608 (2001).
[CrossRef]

Mihalache, D.

D. Mihalache, N. Truta, and L. C. Crasovan, “Painleve analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term,” Phys. Rev. E 56, 1064–1070 (1997).
[CrossRef]

D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
[CrossRef]

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

D. Mihalache, F. Lederer, and D.-M. Baboiu, “Two-parameter family of exact solutions of the nonlinear Schrödinger equation describing optical soliton propagation,” Phys. Rev. A 47, 3285–3290 (1993).
[CrossRef] [PubMed]

D. Mihalache and N. C. Panoiu, “Analytic method for solving the nonlinear Schrödinger equation describing pulse propagation in dispersive optic fibres,” J. Phys. A: Math. Gen. 26, 2679–2697 (1993).
[CrossRef]

Mitzkevich, N. V.

N. N. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

Moldoveanu, F.

D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
[CrossRef]

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

Nakkeeran, K.

K. Porsezian and K. Nakkeeran, “Optical solitons in presence of Kerr dispersion and self-frequency shift,” Phys. Rev. Lett. 76, 3955–3958 (1996).
[CrossRef] [PubMed]

Palacios, S. L.

S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, and R. D. Crespo, “Dark solitary waves in the nonlinear Schrödinger equation with third-order dispersion, self-steepening, and self-frequency shift,” Phys. Rev. E 60, R45–R47 (1999).
[CrossRef]

Panoiu, N. C.

D. Mihalache and N. C. Panoiu, “Analytic method for solving the nonlinear Schrödinger equation describing pulse propagation in dispersive optic fibres,” J. Phys. A: Math. Gen. 26, 2679–2697 (1993).
[CrossRef]

Panoiu, N.-C.

D. Mihalache, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “The Riemann problem method for solving a perturbed nonlinear Schrödinger equation describing pulse propagation in optic fibers,” J. Phys. A: Math. Gen. 27, 6177–6189 (1994).
[CrossRef]

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

Park, Q. H.

Q. H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. 82, 4432–4435 (1999).
[CrossRef]

Porsezian, K.

A. Mahalingam and K. Porsezian, “Propagation of dark solitons with higher-order effects in optical fibers,” Phys. Rev. E 64, 046608 (2001).
[CrossRef]

K. Porsezian and K. Nakkeeran, “Optical solitons in presence of Kerr dispersion and self-frequency shift,” Phys. Rev. Lett. 76, 3955–3958 (1996).
[CrossRef] [PubMed]

Radhakrishnam, R.

R. Radhakrishnam and M. Lakshmanan, “Exact soliton solutions to coupled nonlinear Schrödinger equations with higher-order effects,” Phys. Rev. E 54, 2949–2955 (1996).
[CrossRef]

Sasa, N.

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Satsuma, J.

N. Sasa and J. Satsuma, “New-type soliton solution for a higher-order nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 60, 409–417 (1991).
[CrossRef]

Scott, T. C.

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical solitary waves in the higher-order nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 448–451 (1997).
[CrossRef]

Shin, H. J.

Q. H. Park and H. J. Shin, “Parametric control of soliton light traffic by cw traffic light,” Phys. Rev. Lett. 82, 4432–4435 (1999).
[CrossRef]

Spatscheck, K. H.

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

Su, D. C.

Z. H. Li, G. S. Zhou, and D. C. Su, “N-soliton solution in the higher order nonlinear Schrödinger equation,” in Fiber Optic Components and Optical Communication II, S. Jian, F. F. Tong, and R. Maerz, eds., Proc. SPIE 3552, 226–231 (1998).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Tian, H. P.

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Torner, L.

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

Truta, N.

D. Mihalache, N. Truta, and L. C. Crasovan, “Painleve analysis and bright solitary waves of the higher-order nonlinear Schrödinger equation containing third-order dispersion and self-steepening term,” Phys. Rev. E 56, 1064–1070 (1997).
[CrossRef]

D. Mihalache, L. Torner, F. Moldoveanu, N.-C. Panoiu, and N. Truta, “Inverse scattering approach to femtosecond solitons in monomode optical fibers,” Phys. Rev. E 48, 4699–4709 (1993).
[CrossRef]

Xu, Z. Y.

Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[CrossRef]

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

Zhou, G. S.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Z. Y. Xu, L. Li, Z. H. Li, and G. S. Zhou, “Modulation instability and solitons on a cw background in an optical fiber with higher-order effects,” Phys. Rev. E 67, 026603 (2003).
[CrossRef]

L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatscheck, “Gray optical dips in subpicosecond regime,” Phys. Rev. E 66, 046616 (2002).
[CrossRef]

Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, “New types of solitary wave solution for the higher nonlinear Schrödinger equation,” Phys. Rev. Lett. 84, 4096–4099 (2000).
[CrossRef] [PubMed]

Z. H. Li, G. S. Zhou, and D. C. Su, “N-soliton solution in the higher order nonlinear Schrödinger equation,” in Fiber Optic Components and Optical Communication II, S. Jian, F. F. Tong, and R. Maerz, eds., Proc. SPIE 3552, 226–231 (1998).
[CrossRef]

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersion dielectric fibers. 1. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

IEEE J. Quantum Electron.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[CrossRef]

N. N. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in an optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

J. Math. Phys.

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Figures (5)

Fig. 1
Fig. 1

(a) Evolution of the numerical solution of the initial value problem associated with Eq. (2) with initial condition (8) under the integrability conditions; (b) Evolution of the numerical solution under conditions deviating from those ensuring the integrability of Eq. (2). Parameters are as follows: μ=1, Ac=0.9, As=1, ω=0.2, t0=6, φ0=0, α1=0.5, α2=1, α3=0.05, α4=0.3, and α5=-0.3.

Fig. 2
Fig. 2

Plots showing the exact solution (7) with and without the higher-order effects at z=0, z=4.2, z=8.4, z=11.2, and z=14, respectively. The parameters are α3=0.05, α4=0.3, α5=-0.3 (solid curves); and α3=α4=α5=0 (dashed curves). The other parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Evolution of the solution (11) over tens of normalized propagation lengths for (a) Ac0 and (b) Ac=0. The parameters are the same as in Fig. 1 except for (a) Ac=0.3 and (b) Ac=0.

Fig. 4
Fig. 4

Evolution of the soliton’s peak position of the single-soliton solution q1-sol (solid line) and the soliton solution (11) (dashed line). The parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

(a) Evolution of the maximum intensity (dashed curve), the minimum intensity (dotted curve), and the cw background intensity (solid curve); (b) The location of the maximum intensity (solid curve) and the minimum intensity (dotted curve) in the time–propagation-distance plane. The parameters are the same as in Fig. 3(a).

Equations (38)

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qz=i(α1qtt+α2|q|2q)+α3qttt+α4(|q|2q)t+α5q(|q|2)t,
qz=i(α1qtt+α2|q|2q)+α3qttt+α4|q|2qt,
ψt=Uψ,ψz=Vψ,
U=λJ+μP,
V=4α3λ3J+i2α1λ2J+4µα3λ2P+i2µλQ1+μQ2,
J=100-1,P=0-qq¯0,
Q1=-iμα3|q|2-α1q+iα3qtα1q¯+iα3q¯tiμα3|q|2,
Q2=AB-B¯-A,
q1=q-2μ(λ+λ¯)ψ1ψ¯2ψTψ¯,
qN=q-2μm=1N(λm+λ¯m)ψ1[m, λm]ψ¯2[m, λm]ψ[m, λm]Tψ¯[m, λm],
ψ[m, λm]=(λm-S[m-1])(λm-S[1])ψ[1, λm],
Ssl[j]=-λ¯jδsl+(λj+λ¯j)ψs[j, λj]ψ¯l[j, λj]ψ[j, λj]Tψ¯[j, λj],
q1=exp(iφ)Ac+As a cosh θ1+cos φ1cosh θ1+a cos φ1+iAs b sinh θ1+c sin φ1cosh θ1+a cos φ1,
θ1=i(M1-M¯1)t-(M1β1+M¯1β¯1)z-t0,
φ1=(M1+M¯1)t+i(M1β1-M¯1β¯1)z-φ0,
φ=ωct+κz,
a=μAc(L1+L¯1)/(μ2Ac2+|L1|2),
b=iμAc(L1-L¯1)/(μ2Ac2+|L1|2),
c=(μ2Ac2-|L1|2)/(μ2Ac2+|L1|2),
L1=-μAs/2-i(ωc-ωs)/2-iM1,
M1=12[(ωc-ωs-iμAs)2+4μ2Ac2]1/2,
q1=exp(iφ)Ac+As×2Ac cos φ1-As  cosh θ1+iM1R  sinh θ12Ac  cosh θ1-As cos φ1,
q1(0, t)[ρ+χ cos φ1]exp(iϖt),
q=[Ac+A(t, z)]exp(iα2Ac2z),
Az=iα1Att+iα2Ac2(A+A¯)+α3Attt+(2α4+α5)Ac2At+(α4+α5)Ac2A¯t.
A=u cos(Kz-Ωt)+iv sin(Kz-Ωt),
[K-α3Ω3+(3α4+2α5)Ac2Ω]u+α1Ω2v=0,
(2α2Ac2-α1Ω2)u+(-K+α3Ω3-α4Ac2Ω)v=0.
K=α3Ω3-(2α4+α5)Ac2Ω±|α1Ω|Ω2-Ωc2,
Ωc2[2α1α2-(α4+α5)2Ac2]Ac2α12,
(α4+α5)2Ac2<2α1α2.
q1=exp[i(ϖt+κz)]×M1I M1I cos φ1+iAs sin φ1As  cosh θ1-2Ac cos φ1-Ac,
|q1|2=Ac2+AsM1I2As-2Ac cos φ1,
|q1|2=Ac2-Ac2M1I2  cos2 φ1As2-4Ac2  cos2 φ1.
q1=exp(iφ)(-Ac+iM1I  sech θ1)
-+[|q(t, z)|2-|q(±, z)|2]dt=2M1Iμ.
-+|q(t, z)-q(±, z)|2dt=2M1Iμ(1+AcI cos φ1),
I=4arctanhAs+2Ac cos φ1As-2Ac cos φ11/2(As2-4Ac2  cos2 φ1)1/2,

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