Abstract

We propose a scheme to generate superluminal optical solitons in a four-level atomic system with two control fields via an active Raman gain. We derive a modified nonlinear Schrödinger equation with high-order corrections contributed from linear and differential absorption, nonlinear dispersion, and delay response of nonlinear refractive index of the system. We predict various optical solitons in different regimes of system parameters, and show that these optical solitons have superluminal propagating velocity and very low generation power.

© 2011 OSA

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References

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  1. A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
  3. G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).
  4. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
    [Crossref]
  5. Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
    [Crossref] [PubMed]
  6. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
    [Crossref]
  7. C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
    [Crossref]
  8. W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
    [Crossref]
  9. L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
    [Crossref] [PubMed]
  10. K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
    [Crossref]
  11. R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
    [Crossref] [PubMed]
  12. A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
    [Crossref] [PubMed]
  13. R. Y. Chiao and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345.
    [Crossref]
  14. L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
    [Crossref]
  15. A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
    [Crossref]
  16. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
    [Crossref] [PubMed]
  17. A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
    [Crossref]
  18. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
    [Crossref] [PubMed]
  19. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
    [Crossref]
  20. M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
    [Crossref]
  21. R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
    [Crossref]
  22. K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
    [Crossref]
  23. L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
    [Crossref]
  24. E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
    [Crossref]
  25. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
    [Crossref]
  26. A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
    [Crossref]
  27. M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
    [Crossref]
  28. J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006).
    [Crossref] [PubMed]
  29. K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
    [Crossref]
  30. G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
    [Crossref]
  31. C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010).
    [Crossref]
  32. A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).
  33. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein.
    [Crossref]
  34. M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation,” Phys. Rev. Lett. 78, 448 (1997).
    [Crossref]
  35. K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
    [Crossref]
  36. S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
    [Crossref]

2010 (2)

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010).
[Crossref]

2008 (2)

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

2007 (2)

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

2006 (3)

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

J. Zhang, G. Hernandez, and Y. Zhu, “Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes,” Opt. Lett. 31, 2598 (2006).
[Crossref] [PubMed]

2005 (2)

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
[Crossref]

2004 (4)

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

2003 (7)

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

2002 (1)

S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein.
[Crossref]

2001 (2)

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

2000 (1)

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

1999 (1)

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

1998 (1)

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

1997 (1)

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation,” Phys. Rev. Lett. 78, 448 (1997).
[Crossref]

1994 (1)

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

1993 (1)

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

Agrawal, G. P.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).

Akulshin, A. M.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Aranson, S.

S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein.
[Crossref]

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 (1997).
[Crossref]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Chen, A.-X.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Chiao, R. Y.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref] [PubMed]

R. Y. Chiao and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345.
[Crossref]

Cimmino, A.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Crespo, R. D.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

Dasgupta, S.

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

Deng, L.

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Dogariu, A.

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

Fernndez-Dlaz, J. M.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
[Crossref]

Gauthier, D. J.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

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 (1997).
[Crossref]

Ghulghazaryan, R. G.

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

Guinea, A.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

Hagley, E. W.

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

Hang, C.

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

Hannaford, P.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Hasegawa, A.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).

Hernandez, G.

Huang, G.

C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
[Crossref]

Janowicz, M.

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

Jeffery, A.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Jiang, K.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Jiang, K. J.

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

Kawahawa, T.

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

Kim, J. B.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kim, K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kim, S. K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

Kramer, L.

S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein.
[Crossref]

Kuzmich, A.

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Lee, C.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Lee, R.-K.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Lezama, A.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

Lin, Q.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Liu, N.-H.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Mahalingam, A.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

Malakyan, Y. P.

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
[Crossref]

Matsumoto, M.

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).

Mikhailov, E. E.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Milonni, P. W.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

Moon, H. S.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

Mostowski, J.

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

Nakkeeran, K.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

Neifield, M. A.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Novikova, I.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Opat, G. I.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Palacios, S. L.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

Payne, M. G.

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Pogariu, P.

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Porsezian, K.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

Sautenkov, V. A.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[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 (1997).
[Crossref]

Shanmugha Sundaram, P.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

Si, L.-G.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Sidorov, A. I.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

R. Y. Chiao and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345.
[Crossref]

Stenner, M. D.

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Wang, L. J.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

Wang, L.-G.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Welch, G. R.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

Wu, Y.

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Yang, W.-X.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Yang, X.

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

Zhang, J.

Zhu, C.

C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010).
[Crossref]

Zhu, S.-Y.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

Zhu, Y.

European Phys. J. D (1)

C. Zhu and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” European Phys. J. D 56, 231 (2010).
[Crossref]

Nature (London) (2)

L. J. Wang, A. Kuzmich, and P. Pogariu, “Superluminal solitons in a Lambda-type atomic system with two-folded levels,” Nature (London) 406, 277 (2000).
[Crossref]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature (London) 425, 695 (2003).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (15)

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, “Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas,” Phys. Rev. A 69, 063808 (2004).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, “Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light,” Phys. Rev. A 73, 033806 (2006).
[Crossref]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, “Light propagation in an atomic medium with steep and sign-reversible dispersion,” Phys. Rev. A 67, 011801(R) (2003).
[Crossref]

K. J. Jiang, L. Deng, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 76, 033819 (2007).
[Crossref]

G. S. Agarwal and S. Dasgupta, “Superluminal propagation via coherent manipulation of the Raman gain process,” Phys. Rev. A 70, 023802 (2004).
[Crossref]

M. D. Stenner and D. J. Gauthier, “Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation,” Phys. Rev. A 67, 063801 (2003).
[Crossref]

R. G. Ghulghazaryan and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
[Crossref]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, “Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line,” Phys. Rev. A 68, 013810 (2003).
[Crossref]

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001).
[Crossref]

C. Hang and G. Huang, “Weak-light ultraslow vector solitons via electromagnetically induced transparency,” Phys. Rev. A 77, 033830 (2008).
[Crossref]

W.-X. Yang, A.-X. Chen, L.-G. Si, K. Jiang, X. Yang, and R.-K. Lee, “Three coupled ultraslow temporal solitons in a five-level tripod atomic system,” Phys. Rev. A 81, 023814 (2010).
[Crossref]

K. J. Jiang, L. Deng, E. W. Hagley, and M. G. Payne, “Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
[Crossref]

R. Y. Chiao, “Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations,” Phys. Rev. A 48, R34 (1993).
[Crossref] [PubMed]

A. M. Steinberg and R. Y. Chiao, “Dispersionless, highly superluminal propagation in a medium with a gain doublet,” Phys. Rev. A 49, 2071 (1994).
[Crossref] [PubMed]

Phys. Rev. E (3)

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E 68, 066606 (2003).
[Crossref]

M. Janowicz and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E 73, 046613 (2006).
[Crossref]

G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E 72, 016617 (2005).
[Crossref]

Phys. Rev. E. (1)

S. L. Palacios, A. Guinea, J. M. Fernndez-Dlaz, 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 (1999).
[Crossref]

Phys. Rev. Lett. (5)

M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation,” Phys. Rev. Lett. 78, 448 (1997).
[Crossref]

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schrödinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
[Crossref]

L. Deng and M. G. Payne, “Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium,” Phys. Rev. Lett. 98, 253902 (2007).
[Crossref] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925 (2001).
[Crossref] [PubMed]

Y. Wu and L. Deng, “Ultraslow Optical Solitons in a Cold Four-State Medium,” Phys. Rev. Lett. 93, 143904 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005), and references therein.
[Crossref]

S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99 (2002), and references therein.
[Crossref]

Science (1)

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a roomtemperature solid,” Science 301, 200 (2003).
[Crossref] [PubMed]

Other (5)

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

A. Hasegawa and M. Matsumoto, Optical Solitons in Fibers (Springrer, Berlin, 2003).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Elsevier Pte Ltd, Singapore, 2009).

R. Y. Chiao and A. M. Steinberg, Progress in optics, edited by E. Wolf (Elsevier, Amsterdam, 1997), p. 345.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

(Color online) Energy-level diagram and excitation scheme of Λ-type four state atomic system with active Raman gain. |j〉 (j = 1, 2, 3, 4) are bare atomic states. Δ4 is one-photon detuning. Ωp is the half Rabi frequency of weak, pulsed probe field. Ωc and Ωb are half Rabi frequencies of two strong, continuous control fields.

Fig. 2
Fig. 2

(Color online) (a) Negative imaginary part (gain) −Im(K) and (b) real part Re(K) as functions of ω. Solid and dashed lines correspond to Ωb = 1.0 × 104 s−1 and Ωb = 1.0 × 106 s−1, respectively.

Fig. 3
Fig. 3

(Color online) Values of coefficients g0, g4 (panel (a)), and g1, g2, g3 (panel (b)) of Eq. (19) as function of the pulse length τ0. Both panels (a) and (b) are plotted by the parameters given by γ2 = 1.5 × 104s−1, γ3 = 5. 5 × 104s−1, γ4 = 1.8 × 108s−1, Ωc = 1. 5 × 107s−1, Ωb = 1.0 × 106s−1, Δ2 = 0.1 × 106s−1, Δ3 = 0.5 × 106s−1, Δ4 = −1.0 × 109s−1, and κ34 = 1 × 109cm−1s−1.

Fig. 4
Fig. 4

(Color online) (a) The wave shape of |Ωp/U0|2 versus time t and propagating distance z based on Eqs. (1a)(1d). The initial condition (black solid line) is given by Eq. (23) with τ0 = 1.5 × 10−6s. The (red) dot-dashed line is the soliton profile at the propagation distance z = 2LD. The (green) dotted line is the result for τ0 = 3.5 × 10−6s. The (blue) dashed line the soliton profile at the propagation distance z = 4LD. (b) Collision between two solitons with the initial condition u(0,σ) = 1.0sech(σ – 3.0) exp(–i1.2σ + iπ) + 1.0sech(σ + 5.0) exp(i1.2σ). (c) Collision between two solitons with the initial condition u(0, σ) = 1.0sech(σ − 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ). (d) Collision between two solitons with larger amplitude amplitudes. The initial condition is u(0,σ) = 1.5sech(σ – 3.0) exp(−i1.2σ) + 1.0sech(σ + 5.0) exp(i1.2σ).

Equations (44)

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( i t + d 2 ) A 2 + Ω b * A 3 = 0 ,
( i t + d 3 ) A 3 + Ω p * A 4 + Ω b A 2 = 0 ,
( i t + d 4 ) A 4 + Ω c A 1 + Ω p A 3 = 0 ,
i ( z + 1 c t ) Ω p + κ 34 A 4 A 3 * = 0 ,
K ( ω ) = ω c κ 34 ( ω + d 2 ) | Ω c | 2 | d 4 | 2 D | a 1 | 2 ,
A j = n = 0 ɛ n a j ( n ) , Ω p * = n = 1 ɛ n Ω p ( n ) , ( j = 1 , 2 , 3 )
( i t 0 + d 2 ) a 2 ( l ) + Ω b * a 3 ( l ) = M ( l ) ,
( i t 0 + d 3 ) a 3 ( l ) + Ω p ( l ) a 4 ( 0 ) + Ω b a 2 ( l ) = N ( l )
( i t 0 + d 4 * ) a 4 ( l ) * + Ω s ( l ) * a 1 ( l ) * = Q ( l ) ,
i ( z 0 + 1 c t 0 ) Ω p ( l ) + κ 34 a 3 ( l ) a 4 ( 1 ) * = R ( l ) .
a 2 ( l ) = 1 Ω b [ N ( l ) ( i t 0 + d 3 ) a 3 ( l ) Ω p ( l ) a 4 ( 0 ) ] ,
a 3 ( l ) = 1 κ 34 a 4 ( 0 ) * [ R ( l ) + i ( z 0 + 1 c t 0 ) Ω p ( l ) ] ,
L ^ Ω p ( l ) = S ( l ) ,
L ^ = i ( z 0 + 1 c t 0 ) [ | Ω b | 2 ( i t 0 + d 2 ) ( i t 0 + d 3 ) ] κ 34 | a 4 ( 0 ) | 2 ( i t 0 + d 2 ) ,
S ( l ) = R ( l ) [ ( i t 0 + d 2 ) ( i t 0 + d 3 ) | Ω b | 2 ] + κ 34 a 4 ( 0 ) * [ Ω b M ( l ) ( i t 0 + d 2 ) N ( l ) ]
Ω p ( 1 ) = F e i θ ,
a 2 ( 1 ) = a 4 ( 0 ) Ω b * D F e i θ ,
a 3 ( 1 ) = a 4 ( 0 ) ( ω + d 2 ) D F e i θ ,
L ^ Ω p ( 2 ) = i D e i θ ( F z 1 + 1 v g F t 1 ) .
i ( F z 1 + 1 v g F t 1 ) = 0
Ω p ( 2 ) = 0 ,
a 2 ( 2 ) = i e i θ Ω b κ 34 a 4 ( 0 ) * [ ( K ω c ) + ( ω + d 3 ) ( 1 v g 1 c ) ] F t 1 ,
a 3 ( 2 ) = i e i θ κ 34 a 4 ( 0 ) * ( 1 v g 1 c ) F t 1 ,
a 4 ( 0 ) a 4 ( 2 ) * + a 4 ( 0 ) * a 4 ( 2 ) = i e α ¯ z 2 | F | 2 | a 4 ( 0 ) | 2 | Ω c | 2 + | d 4 | 2 { | Ω c | 2 ( | ω + d 2 D | 2 + | Ω b D | 2 ) | d 4 | 2 [ ( ω + d 2 D ) * d 4 + c . c . ] } ,
L ^ Ω p ( 3 ) = D [ i F z 2 K 2 2 2 F t 1 2 W | F | 2 F ] e i θ
W = i κ 34 ( ω + d 2 ) e α ¯ z 2 | a 4 ( 0 ) | 2 | Ω c | 2 + | d 4 | 2 { | Ω c | 2 ( | ω + d 2 D | 2 + | Ω b D | 2 ) | d 4 | 2 [ ( ω + d 2 D ) * d 4 + c . c . ] } .
i F z 2 K 2 2 2 F t 1 2 W | F | 2 F = 0 ,
a 1 ( 3 ) = i ( α 1 F * F t 1 α 1 * F F * t 1 ) e α ¯ z 2 ,
a 2 ( 3 ) = ( α 2 ( 1 ) 2 F t 1 2 + α 2 ( 2 ) | F | 2 F ) e i θ ,
a 3 ( 3 ) = ( α 3 ( 1 ) 2 F t 1 2 + α 3 ( 2 ) | F | 2 F ) e i θ ,
Ω p ( 3 ) = 0 ,
i F z 3 i K 3 6 3 F t 1 3 i β 1 e α ¯ z 2 t 1 ( | F | 2 F ) + i β 2 e α ¯ z 2 F t 1 ( | F | 2 ) = 0 ,
i U z + i α 2 U K 2 2 2 U τ 2 i K 3 6 3 U τ 3 W | U | 2 U i β 1 τ ( | U | 2 U ) + i β 2 U τ ( | U | 2 ) = 0 ,
i u s + 2 u σ 2 + 2 | u | 2 u = i ( g 0 u + g 1 ( | u | 2 u ) σ + g 2 u | u | 2 σ + g 3 3 u σ 3 ) + g 4 u σ .
i u s + 2 u σ 2 + 2 u | u | 2 = i g 0 u ,
Ω p = e g 0 z / L D τ 0 K ˜ 2 W ˜ sech [ e g 0 z / L D τ 0 ( t z V ˜ g ) ] exp [ i K ˜ 0 z i 1 e 2 g 0 z / L D 4 g 0 ] .
i u s + 2 u σ 2 + 2 u | u | 2 = i [ g 1 ( | u | 2 u ) σ + g 2 u ( | u | 2 ) σ + g 3 3 u σ 3 ] .
Ω p = U 0 3 ( β + 3 q 2 2 q ) g 3 2 ( 3 c 1 + c 2 ) sech [ β + 3 q 2 2 q g 3 ( t z / V ˜ g τ 0 + β z 2 g 3 L D ) ] × exp ( i [ q 3 q 2 + ( β + 3 q 2 2 q ) ( 1 3 q ) ] z 2 g 3 2 L D i q g 3 t z / V ˜ g τ 0 + i K ˜ 0 z ) ,
V ˜ g = 2.24 × 10 5 c ,
V ˜ g H = 2.379 × 10 6 c
P ¯ = P ¯ max , 1 sech 2 [ 1 τ 0 ( t z V ˜ g ) ] , ( for the soliton ( 21 ) ) ,
P ¯ = P ¯ max , 2 × sech 2 [ β + 3 q 2 2 q d 3 τ 0 ( t z V ˜ g H ) ] , ( for the soliton ( 23 ) )
P ¯ max , 1 2 ɛ 0 c n p S 0 ( h ¯ | μ 34 | ) 2 1 τ 0 2 K ˜ 2 W ˜ ,
P ¯ max , 2 ɛ 0 c n p S 0 ( h ¯ | μ 34 | ) 2 6 ( β + 3 q 2 2 q ) g 3 2 τ 0 2 ( 3 c 1 + 2 c 2 ) K ˜ 2 W ˜ ,

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