Abstract

The adiabatic dynamics of solitons under the action of third-order dispersion (TOD), the Raman effect, and self-steepening is studied. Using equations that describe the evolution of the pulse parameters, it is shown that the interplay between these effects results in nontrivial pulse dynamics. It is found that positive TOD slows down the self-frequency shift. The theory also describes the eventual suppression of the self-frequency shift in fibers with negative TOD that was recently observed in experiments and described theoretically. The relations of our results to supercontinuum generation are discussed.

© 2006 Optical Society of America

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  2. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, 1997).
  3. A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer, 1999).
  4. See, for example, R.R.Alfano, ed., The Supercontinuum Laser Source (Springer-Verlag, 1989).
  5. See, for example, the special issue on supercontinuum generation, Appl. Phys. B 77, A. Zheltikov, ed. (2003).
  6. J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm," Opt. Lett. 25, 25-27 (2000).
    [CrossRef]
  7. A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef] [PubMed]
  8. J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton, and S. Coen, "Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping," J. Opt. Soc. Am. B 19, 765-771 (2002).
    [CrossRef]
  9. A. Fuerbach, P. Steinvurzel, J. A. Bolger, and B. J. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005).
    [CrossRef] [PubMed]
  10. R. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, "Nonlinear pulse-propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers," Opt. Lett. 11, 464-466 (1986).
    [CrossRef] [PubMed]
  11. F. M. Mitschke and L. F. Mollenauer, "Discovery of the soliton self-frequency shift," Opt. Lett. 11, 659-661 (1986).
    [CrossRef] [PubMed]
  12. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662-664 (1986).
    [CrossRef] [PubMed]
  13. Yu. Kodama and A. Hasegawa, "Nonlinear pulse-propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
    [CrossRef]
  14. K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman-scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
    [CrossRef]
  15. J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
    [CrossRef]
  16. T. P. Horikis and J. N. Elgin, "Soliton radiation in an optical fiber," J. Opt. Soc. Am. B 18, 913-918 (2001).
    [CrossRef]
  17. W. Zhao and E. Bourkoff, "Femtosecond pulse-propagation in optical fibers: higher-order effects," IEEE J. Quantum Electron. 24, 365-372 (1987).
    [CrossRef]
  18. D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
    [CrossRef]
  19. A. Ankiewicz, "Simplified description of soliton perturbation and interaction using averaged complex potentials," J. Nonlinear Opt. Phys. Mater. 4, 857-870 (1995).
    [CrossRef]
  20. M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
    [CrossRef]
  21. V. I. Karpman, "Radiation of solitons described by a high-order cubic nonlinear Schrödinger equation," Phys. Rev. E 62, 5678-5687 (2000).
    [CrossRef]
  22. D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
    [CrossRef] [PubMed]
  23. F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
    [CrossRef]
  24. A. I. Maimistov, "Evolution of single waves close to solitons of Schrödinger nonlinear equation," J. Exp. Theor. Phys. 77, 727 (1993) A. I. Maimistov,[Zh. Eksp. Teor. Fiz. 104, 3620-3629 (1993)].
  25. E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
    [CrossRef]
  26. W. L. Kath and N. F. Smith, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
    [CrossRef]
  27. D. J. Kaup and A. C. Newell, "Exact solution for a derivative non-linear Schrödinger equation," J. Math. Phys. 19, 798-801 (1978).
    [CrossRef]
  28. See, for example, B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. Vol. 43, (Elsevier, 2002), pp. 71-193, and references therein.
    [CrossRef]
  29. F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
    [CrossRef]
  30. H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).
  31. J. Satsuma and N. Yajima, "Initial value-problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Prog. Theor. Phys. Suppl. 55, 284-306 (1974).
    [CrossRef]
  32. A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russell, "Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: experiment and modelling," Opt. Express 12, 6498-6507 (2004).
    [CrossRef] [PubMed]
  33. K. M. Hilligsøe, T. V. Andersen, H. N. Paulsen, C. K. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. P. Hansen, and J. J. Larsen, "Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths," Opt. Express 12, 1045-1054 (2004).
    [CrossRef] [PubMed]
  34. M. H. Frosz, P. Falk, and O. Bang, "The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength," Opt. Express 13, 6181-6192 (2005).
    [CrossRef] [PubMed]

2006 (1)

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
[CrossRef]

2005 (2)

2004 (3)

2003 (2)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

See, for example, the special issue on supercontinuum generation, Appl. Phys. B 77, A. Zheltikov, ed. (2003).

2002 (2)

2001 (2)

T. P. Horikis and J. N. Elgin, "Soliton radiation in an optical fiber," J. Opt. Soc. Am. B 18, 913-918 (2001).
[CrossRef]

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

2000 (2)

V. I. Karpman, "Radiation of solitons described by a high-order cubic nonlinear Schrödinger equation," Phys. Rev. E 62, 5678-5687 (2000).
[CrossRef]

J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm," Opt. Lett. 25, 25-27 (2000).
[CrossRef]

1998 (1)

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

1997 (1)

M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
[CrossRef]

1995 (4)

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

A. Ankiewicz, "Simplified description of soliton perturbation and interaction using averaged complex potentials," J. Nonlinear Opt. Phys. Mater. 4, 857-870 (1995).
[CrossRef]

J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
[CrossRef]

W. L. Kath and N. F. Smith, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

1993 (1)

A. I. Maimistov, "Evolution of single waves close to solitons of Schrödinger nonlinear equation," J. Exp. Theor. Phys. 77, 727 (1993) A. I. Maimistov,[Zh. Eksp. Teor. Fiz. 104, 3620-3629 (1993)].

1989 (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman-scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

1987 (2)

W. Zhao and E. Bourkoff, "Femtosecond pulse-propagation in optical fibers: higher-order effects," IEEE J. Quantum Electron. 24, 365-372 (1987).
[CrossRef]

Yu. Kodama and A. Hasegawa, "Nonlinear pulse-propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

1986 (3)

1978 (1)

D. J. Kaup and A. C. Newell, "Exact solution for a derivative non-linear Schrödinger equation," J. Math. Phys. 19, 798-801 (1978).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, "Initial value-problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Prog. Theor. Phys. Suppl. 55, 284-306 (1974).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

Agrawal, G. P.

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

Akhmediev, N.

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
[CrossRef]

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

Andersen, T. V.

Ankiewicz, A.

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
[CrossRef]

A. Ankiewicz, "Simplified description of soliton perturbation and interaction using averaged complex potentials," J. Nonlinear Opt. Phys. Mater. 4, 857-870 (1995).
[CrossRef]

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

Bang, O.

Basharov, A. M.

A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer, 1999).

Bella, I.

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

Biancalana, F.

Blow, K. J.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman-scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Bolger, J. A.

Bourkoff, E.

W. Zhao and E. Bourkoff, "Femtosecond pulse-propagation in optical fibers: higher-order effects," IEEE J. Quantum Electron. 24, 365-372 (1987).
[CrossRef]

Brabec, T.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
[CrossRef]

Caputo, J. G.

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

Chen, H. H.

Coen, S.

Dudley, J. M.

Efimov, A.

Eggleton, B. J.

Elgin, J. N.

T. P. Horikis and J. N. Elgin, "Soliton radiation in an optical fiber," J. Opt. Soc. Am. B 18, 913-918 (2001).
[CrossRef]

J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
[CrossRef]

Falk, P.

Frantzeskakis, D. J.

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

Frosz, M. H.

Fuerbach, A.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

Golles, M.

M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
[CrossRef]

Gordon, J. P.

Grossard, N.

Hansen, K. P.

Hasegawa, A.

Yu. Kodama and A. Hasegawa, "Nonlinear pulse-propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

Herrmann, J.

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Hilligsøe, K. M.

Hizanidis, K

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

Horikis, T. P.

Husakou, A. V.

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Joly, N. Y.

Karpman, V. I.

V. I. Karpman, "Radiation of solitons described by a high-order cubic nonlinear Schrödinger equation," Phys. Rev. E 62, 5678-5687 (2000).
[CrossRef]

Kath, W. L.

W. L. Kath and N. F. Smith, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

Kaup, D. J.

D. J. Kaup and A. C. Newell, "Exact solution for a derivative non-linear Schrödinger equation," J. Math. Phys. 19, 798-801 (1978).
[CrossRef]

Keiding, S.

Kelly, S. M. J.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
[CrossRef]

Knight, J. C.

Kodama, Yu.

Yu. Kodama and A. Hasegawa, "Nonlinear pulse-propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

Kristiansen, R.

Larsen, J. J.

Lederer, F.

M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
[CrossRef]

Lee, Y. C.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Maillotte, H.

Maimistov, A. I.

A. I. Maimistov, "Evolution of single waves close to solitons of Schrödinger nonlinear equation," J. Exp. Theor. Phys. 77, 727 (1993) A. I. Maimistov,[Zh. Eksp. Teor. Fiz. 104, 3620-3629 (1993)].

A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer, 1999).

Malomed, B. A.

See, for example, B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. Vol. 43, (Elsevier, 2002), pp. 71-193, and references therein.
[CrossRef]

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

Mølmer, K.

Newell, A. C.

D. J. Kaup and A. C. Newell, "Exact solution for a derivative non-linear Schrödinger equation," J. Math. Phys. 19, 798-801 (1978).
[CrossRef]

Nielsen, C. K.

Omenetto, F. G.

Paulsen, H. N.

Provino, L.

Ranka, J. K.

Russell, P. S. J.

Russell, P. St. J.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Satsuma, J.

J. Satsuma and N. Yajima, "Initial value-problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Prog. Theor. Phys. Suppl. 55, 284-306 (1974).
[CrossRef]

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

A. Efimov, A. J. Taylor, F. G. Omenetto, A. V. Yulin, N. Y. Joly, F. Biancalana, D. V. Skryabin, J. C. Knight, and P. S. J. Russell, "Time-spectrally-resolved ultrafast nonlinear dynamics in small-core photonic crystal fibers: experiment and modelling," Opt. Express 12, 6498-6507 (2004).
[CrossRef] [PubMed]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Smith, N. F.

W. L. Kath and N. F. Smith, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

Steinvurzel, P.

Stentz, A. J.

Taylor, A. J.

Tombras, G. S.

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

Tsoy, E. N.

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
[CrossRef]

Uzunov, I. M.

M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
[CrossRef]

Wai, R. K. A.

Windeler, R. S.

Wood, D.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman-scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, "Initial value-problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Prog. Theor. Phys. Suppl. 55, 284-306 (1974).
[CrossRef]

Yulin, A. V.

Zhao, W.

W. Zhao and E. Bourkoff, "Femtosecond pulse-propagation in optical fibers: higher-order effects," IEEE J. Quantum Electron. 24, 365-372 (1987).
[CrossRef]

Appl. Phys. B (1)

See, for example, the special issue on supercontinuum generation, Appl. Phys. B 77, A. Zheltikov, ed. (2003).

IEEE J. Quantum Electron. (4)

W. Zhao and E. Bourkoff, "Femtosecond pulse-propagation in optical fibers: higher-order effects," IEEE J. Quantum Electron. 24, 365-372 (1987).
[CrossRef]

D. J. Frantzeskakis, K Hizanidis, G. S. Tombras, and I. Bella, "Nonlinear dynamics of femtosecond optical solitary wave-propagation at the zero dispersion point," IEEE J. Quantum Electron. 31, 183-189 (1995).
[CrossRef]

Yu. Kodama and A. Hasegawa, "Nonlinear pulse-propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman-scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

J. Exp. Theor. Phys. (1)

A. I. Maimistov, "Evolution of single waves close to solitons of Schrödinger nonlinear equation," J. Exp. Theor. Phys. 77, 727 (1993) A. I. Maimistov,[Zh. Eksp. Teor. Fiz. 104, 3620-3629 (1993)].

J. Math. Phys. (1)

D. J. Kaup and A. C. Newell, "Exact solution for a derivative non-linear Schrödinger equation," J. Math. Phys. 19, 798-801 (1978).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

A. Ankiewicz, "Simplified description of soliton perturbation and interaction using averaged complex potentials," J. Nonlinear Opt. Phys. Mater. 4, 857-870 (1995).
[CrossRef]

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

Opt. Commun. (1)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, "A perturbative theory of soliton propagation in the presence of third-order dispersion," Opt. Commun. 114, 321-328 (1995).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Phys. Lett. A (1)

M. Golles, I. M. Uzunov, and F. Lederer, "Break up of N-soliton bound states due to intrapulse Raman scattering and third order dispersion--an eigenvalue analysis," Phys. Lett. A 231, 195-200 (1997).
[CrossRef]

Phys. Rev. E (5)

V. I. Karpman, "Radiation of solitons described by a high-order cubic nonlinear Schrödinger equation," Phys. Rev. E 62, 5678-5687 (2000).
[CrossRef]

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, "Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation," Phys. Rev. E 73, 036621 (2006).
[CrossRef]

W. L. Kath and N. F. Smith, "Soliton evolution and radiation loss for the nonlinear Schrödinger equation," Phys. Rev. E 51, 1484-1492 (1995).
[CrossRef]

F. Kh. Abdullaev and J. G. Caputo, "Validation of the variational approach for chirped pulses in fibers with periodic dispersion," Phys. Rev. E 58, 6637-6648 (1998).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Prog. Opt. (1)

See, for example, B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. Vol. 43, (Elsevier, 2002), pp. 71-193, and references therein.
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, "Initial value-problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Prog. Theor. Phys. Suppl. 55, 284-306 (1974).
[CrossRef]

Science (1)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancelation in photonic crystal fibers," Science 301, 1705-1708 (2003).
[CrossRef] [PubMed]

Other (5)

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

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

A. I. Maimistov and A. M. Basharov, Nonlinear Optical Waves (Kluwer, 1999).

See, for example, R.R.Alfano, ed., The Supercontinuum Laser Source (Springer-Verlag, 1989).

H. Goldstein, Classical Mechanics (Addison-Wesley, 1980).

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Figures (4)

Fig. 1
Fig. 1

Dependence of (a) soliton width and (b) frequency shift on z for β 3 > 0 . Solid and dashed curves correspond to (a) numerical simulations of Eqs. (27) and Eqs. (34), (b) Eq. (37), while points are from numerical simulations of Eq. (9). The dotted line in (b) is the dependence for β 3 = 0 . Parameters are T R = 0.05 and A 0 = w 0 = 1 .

Fig. 2
Fig. 2

Same as in Fig. 1 but for β 3 < 0 .

Fig. 3
Fig. 3

Pulse evolution found from numerical simulations of Eq. (9). Parameters are β 3 = 0.1 , T R = 0.05 , and A 0 = w 0 = 1 .

Fig. 4
Fig. 4

Dependence of the threshold length on the soliton width. The curve corresponds to Eq. (39), and the points are from the numerical simulations of Eq. (9). Parameters are β 3 = 0.1 , T R = 0.05 , and A 0 = 1 w 0 .

Equations (49)

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i ψ z β 2 2 2 ψ t 2 + γ ψ 2 ψ = i β 3 6 3 ψ t 3 γ [ i T S t ( ψ 2 ψ ) T R ψ ψ 2 t ] R ( ψ ) .
w w c ( n ) [ Φ n ( n ! Φ 1 ) ] 1 ( n 1 ) , n > 1 .
ψ ( t , z ) = A sech ( t t c w ) exp { i [ ϕ + b ( t t c ) + μ ( t t c ) 2 ] } ,
v = v g 1 t c , z v g v g ( 1 + t c , z v g ) ;
Q = ψ 2 d t , P = 1 2 ( ψ ψ t * ψ * ψ t ) d t ,
I 1 = t ψ 2 d t , I 2 = ( t t c ) 2 ψ 2 d t ,
I 3 = ( t t c ) ( ψ * ψ t ψ ψ t * ) d t ,
d Q d z = i ) ( ψ R * ψ * R ) d t ,
d P d z = i ( ψ t R * + ψ t * R ) d t ,
d I 1 d z = i β 2 P + i t ( ψ R * ψ * R ) d t ,
d I 2 d z = i β 2 I 3 + i ( t t c ) 2 ( ψ R * ψ * R ) d t ,
d I 3 d z = 2 P d t c d z i ( 2 β 2 ψ t 2 + γ ψ 4 ) d t + 2 i ( t t c ) ( ψ t R * + ψ t * R ) d t + i ( ψ R * + ψ * R ) d t ,
u ( z ¯ , t ¯ ) = ψ ( z , t ) ψ cs , z ¯ = z z cs , t ¯ = t t cs ,
z cs = t cs 2 β 2 , ψ cs = 1 t cs ( β 2 γ ) 1 2 .
i u z ¯ sgn ( β 2 ) 2 2 u t ¯ 2 + u 2 u = i β ¯ 3 6 3 u t ¯ 3 [ i T ¯ S t ¯ ( u 2 u ) T ¯ R u u 2 t ¯ ] ,
β ¯ 3 = β 3 β 2 t cs , T ¯ S = T S t cs , T ¯ R = T R t cs .
Q z = 0 ,
b z = 2 Q γ 15 w 3 ( 2 T R + 5 T S μ w 2 ) ,
t c , z = β 2 b + β 3 6 w 2 ( 1 + 3 b 2 w 2 + π 2 μ 2 w 4 ) + γ T S Q 2 w ,
w z = 2 β ̂ 2 μ w ,
μ z = 1 π 2 w 4 [ 2 β ̂ 2 ( 1 π 2 μ 2 w 4 ) γ Q w ( 1 T S b ) ] ,
Q = 2 A 2 ( z ) w ( z ) = 2 A 0 2 w 0 = const ,
w z z = 2 β ̂ 2 π 2 [ 2 β ̂ 2 w 3 + γ Q ( 1 T S b ) w 2 ] + β 3 γ Q 15 β ̂ 2 2 w z w 3 ( 5 T S w w z 4 β ̂ 2 T R ) .
w z z = 2 β 2 π 2 ( 2 β 2 w 3 + γ Q w 2 ) U w ,
U = 2 π 2 ( β 2 2 w 2 + γ Q β 2 w ) .
E = w z 2 2 + 2 π 2 ( β 2 2 w 2 + γ Q β 2 w ) .
w s = 2 β 2 γ Q .
A s w s = β 2 γ .
Q z = b z = 0 ,
t c , z = β 2 b + β 3 6 w 2 ( 1 + 3 b 2 w 2 + π 2 μ 2 w 4 ) ,
w z z = 2 β ̂ 2 π 2 ( 2 β ̂ 2 w 3 + γ Q w 2 ) ,
t c , z β 3 6 w s 2 = β 3 γ 2 Q 2 24 β ̂ 2 2 , z .
b z = 4 γ T R Q 15 w 3 .
b z = 8 T R β 2 15 w s 4 .
b z = γ T S Q 3 β ̂ 2 w z w 2 , μ = w z 2 β ̂ 2 w ,
w z z = 2 β ̂ 2 π 2 [ 2 β ̂ 2 w 3 + γ ( 1 T S b ) Q w 2 ] + β 3 γ T S Q 3 β ̂ 2 2 ( w z w ) 2 .
d d z [ β 2 b β 3 b 2 2 γ T S Q 3 w ] = 0 .
b b 0 + γ T S Q 3 β 2 ( 1 w 1 w 0 ) ,
w s = 2 β ̂ 2 γ ( 1 T S b ) Q .
w z z = 2 β ̂ 2 π 2 [ 2 β ̂ 2 w 3 + γ Q w 2 ] 4 15 β 3 γ T R Q β ̂ 2 w z w 3 .
E = w z 2 2 + 2 β ̂ 2 π 2 [ β ̂ 2 w 2 + γ Q w ] .
w s = 2 β ̂ 2 ( γ Q ) ,
b = 1 β 3 [ β 2 + ( β ̂ 2 , 0 4 + 2 15 β 3 T R γ 4 Q 4 z ) 1 4 ] ,
t c , z β 2 b + β 3 2 b 2 ,
z th = 15 2 β ̂ 2 , 0 4 β 3 T R γ 4 Q 4 ,
w s = 2 β ̂ 2 γ ( 1 T S b ) Q
d w s d z 2 ( β 3 β 2 T S ) ( 1 T S b ) 2 γ Q b z + O ( ϵ 2 ) .
d w s d z > 0 , if β 3 > β 2 T S ,
< 0 , if β 3 < β 2 T S .

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