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 Optical Society of America

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  6. G. Huang, L. Deng, and M. G. Payne, “Dynamics of ultraslow optical solitons in a cold three-state atomic system,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 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 Dopplerbroadened three-state single-channel active Raman gain medium,” Phys. Rev. A 77, 045804 (2008).
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    [CrossRef] [PubMed]
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  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]
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  19. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, “The speed of information in a ‘Fast-light’ optical medium,” Nature 425, 695 (2003).
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  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).
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  21. R. G. Ghulghazaryan, and Y. P. Malakyan, “Superluminal optical pulse propagation in nonlinear coherent media,” Phys. Rev. A 67, 063806 (2003).
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  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]
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    [CrossRef]
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    [CrossRef]
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    [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 Stat. Nonlin. Soft Matter Phys. 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,” Eur. Phys. J. D 56, 231 (2010).
    [CrossRef]
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    [CrossRef]
  34. M. Gedalin, T. C. Scott, and Y. B. Band, “Optical Solitary Waves in the Higher Order Nonlinear Schr¨odinger 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 Schr¨odinger Equations,” Phys. Rev. Lett. 80, 1425 (1998).
    [CrossRef]
  36. S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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,” Eur. 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 Dopplerbroadened 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 Stat. Nonlin. Soft Matter Phys. 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 Stat. Nonlin. Soft Matter Phys. 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 (6)

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 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 Stat. Nonlin. Soft Matter Phys. 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 406, 277 (2000).
[CrossRef]

1999 (1)

S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 60, R45 (1999).
[CrossRef]

1998 (1)

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schr¨odinger 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 Schr¨odinger 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]

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]

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 Schr¨odinger 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]

Crespo, R. D.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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 Dopplerbroadened 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 Stat. Nonlin. Soft Matter Phys. 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-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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 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 Schr¨odinger 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-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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 Dopplerbroadened 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]

Hernandez, G.

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]

Huang, G.

C. Zhu, and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” Eur. 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 Stat. Nonlin. Soft Matter Phys. 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 Stat. Nonlin. Soft Matter Phys. 73, 046613 (2006).
[CrossRef]

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 Dopplerbroadened 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]

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]

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. 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 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 Stat. Nonlin. Soft Matter Phys. 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 Stat. Nonlin. Soft Matter Phys. 68, 066606 (2003).
[CrossRef]

Mahalingam, A.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schr¨odinger 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]

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 Stat. Nonlin. Soft Matter Phys. 73, 046613 (2006).
[CrossRef]

Nakkeeran, K.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schr¨odinger 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 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]

Palacios, S. L.

S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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 Dopplerbroadened 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 Stat. Nonlin. Soft Matter Phys. 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 406, 277 (2000).
[CrossRef]

Porsezian, K.

K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam, “Optical Solitons in N-Coupled Higher Order Nonlinear Schr¨odinger 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 Schr¨odinger 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 Schr¨odinger 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]

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]

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 425, 695 (2003).
[CrossRef]

Wang, L. J.

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 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 Stat. Nonlin. Soft Matter Phys. 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.

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]

Zhu, C.

C. Zhu, and G. Huang, “Gain-Assisted Giant Kerr Nonlinearity in a Λ-type System with Two-folded Lower Levels,” Eur. 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 Stat. Nonlin. Soft Matter Phys. 68, 066606 (2003).
[CrossRef]

Zhu, Y.

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]

Eur. Phys. J. D (1)

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

Nature (2)

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

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

Opt. Lett. (1)

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]

Phys. Rev. A (14)

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]

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. 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 Dopplerbroadened 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 Stat. Nonlin. Soft Matter Phys. (3)

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

M. Janowicz, and J. Mostowski, “Superluminal propagation of solitary kinklike waves in amplifying media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73, 046613 (2006).
[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 Stat. Nonlin. Soft Matter Phys. 68, 066606 (2003).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

S. L. Palacios, A. Guinea, J. M. Fernndez-Dłaz, 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 Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 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 Schr¨odinger 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 Schr¨odinger 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 (6)

. 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]

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]

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

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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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