Abstract

We present first ever analytical solutions for shape-preserving pulses in a Kerr nonlinear two-mode fiber doped with 3-level Λ atoms. The two modes are near-resonant with the two transitions of the atomic system. We show the existence of quasi-stable coupled bright-dark pairs if the group velocity dispersion has opposite signs at the two mode frequencies. We demonstrate the remarkable possibility allowed by the fiber dispersion for the existence of a new class of solutions for unequal coupling constants for the two modes. We present the conditions for existence and the analytical form of these solutions in presence of atomic detuning. We confirm numerically the analytical solutions for the spatio-temporal evolution of coupled solitary waves.

© 2008 Optical Society of America

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  1. S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967)
    [CrossRef]
  2. S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
    [CrossRef]
  3. G. L. Lamb, Jr., "Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium," Rev. Mod. Phys. 43, 99-124 (1971).
    [CrossRef]
  4. H. A. Haus, "Physical interpretation of inverse scattering formalism applied to self-induced transparency," Rev. Mod. Phys. 51, 331-339 (1971).
    [CrossRef]
  5. A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
    [CrossRef]
  6. R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
    [CrossRef] [PubMed]
  7. J. H. Eberly, "Transmission of dressed fields in three-level media," Quantum Semiclassic. Opt. 7, 373-384 (1995).
    [CrossRef]
  8. A. Rahman and J. H. Eberly, "Theory of shape-preserving short pulses in inhomogeneously broadened three-leve media," Phys. Rev. A 58, R805-R808 (1998).
    [CrossRef]
  9. J. H. Eberly and V. V. Kozlov, " Wave Equation for Dark Coherence in Three-Level Media," Phys. Rev. Lett. 88, 243604-1-243604-4 (2002).
    [CrossRef]
  10. G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
    [CrossRef]
  11. D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617-1-036617-6 (2002).
    [CrossRef]
  12. S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
    [CrossRef] [PubMed]
  13. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
    [CrossRef]
  14. G. P. Agarwal, Nonlinear Fiber Optics 2nd Ed., (Academic Press, San Diego CA 1995).
  15. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, "Optical solitary waves induced by cross-phase modulation," Opt. Lett. 13, 871-873 (1988).
    [CrossRef] [PubMed]
  16. Q. Yang, J. T. Seo, B. Tabibi, and H. Wang, "Slow Light and Superluminality in Kerr Media without a Pump," Phys. Rev. Lett. 95, 063902-1-063902-4 (2005).
    [CrossRef]
  17. M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
    [CrossRef] [PubMed]
  18. M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
    [CrossRef] [PubMed]
  19. M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
    [CrossRef] [PubMed]
  20. S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, "Resonant Optical Interactions with Molecules Confined in Photonic Band-Gap Fibers," Phys. Rev. Lett. 94, 093902-1-093902-4 (2005).
    [CrossRef]
  21. S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, and B. J. Kirby, "Low-Light-Level Optical Interactions with Rubidium Vapor in a Photonic Band-Gap Fiber," Phys. Rev. Lett. 97, 023603-1-023603-4 (2006).
    [CrossRef]

1999

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

1998

A. Rahman and J. H. Eberly, "Theory of shape-preserving short pulses in inhomogeneously broadened three-leve media," Phys. Rev. A 58, R805-R808 (1998).
[CrossRef]

1997

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

1995

J. H. Eberly, "Transmission of dressed fields in three-level media," Quantum Semiclassic. Opt. 7, 373-384 (1995).
[CrossRef]

1994

R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
[CrossRef] [PubMed]

1992

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef] [PubMed]

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

1991

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

1990

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

1988

1971

G. L. Lamb, Jr., "Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium," Rev. Mod. Phys. 43, 99-124 (1971).
[CrossRef]

H. A. Haus, "Physical interpretation of inverse scattering formalism applied to self-induced transparency," Rev. Mod. Phys. 51, 331-339 (1971).
[CrossRef]

1969

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

1967

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967)
[CrossRef]

Agarwal, G. S.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

Bhasrov, A. M.

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

Eberly, J. H.

A. Rahman and J. H. Eberly, "Theory of shape-preserving short pulses in inhomogeneously broadened three-leve media," Phys. Rev. A 58, R805-R808 (1998).
[CrossRef]

J. H. Eberly, "Transmission of dressed fields in three-level media," Quantum Semiclassic. Opt. 7, 373-384 (1995).
[CrossRef]

R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
[CrossRef] [PubMed]

Elyutin, S. O.

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

Field, J. E.

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef] [PubMed]

Grobe, R.

R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
[CrossRef] [PubMed]

Hahn, E. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967)
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef] [PubMed]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

Haus, H. A.

H. A. Haus, "Physical interpretation of inverse scattering formalism applied to self-induced transparency," Rev. Mod. Phys. 51, 331-339 (1971).
[CrossRef]

Hioe, F. T.

R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
[CrossRef] [PubMed]

Kasapi, A.

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef] [PubMed]

Kimura, Y.

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

Kubota, H.

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Kurokawa, K.

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

Lamb, G. L.

G. L. Lamb, Jr., "Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium," Rev. Mod. Phys. 43, 99-124 (1971).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

McCall, S. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967)
[CrossRef]

Nakazawa, M.

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Rahman, A.

A. Rahman and J. H. Eberly, "Theory of shape-preserving short pulses in inhomogeneously broadened three-leve media," Phys. Rev. A 58, R805-R808 (1998).
[CrossRef]

Sklyarov, M. Y.

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

Stegeman, G. I.

Suzuki, K.

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

Trillo, S.

Vasavada, K. V.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Vemuri, G.

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

Wabnitz, S.

Wright, E. M.

Yamada, E.

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

Nature

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature,  397, 594-598 (1999).
[CrossRef]

Opt. Lett.

Phys. Rep.

A. I. Maimistov, A. M. Bhasrov, S. O. Elyutin, and M. Y. Sklyarov, "Present state of self-induced transparency theory," Phys. Rep. 191, 1-108 (1990).
[CrossRef]

Phys. Rev.

S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969).
[CrossRef]

Phys. Rev. A

A. Rahman and J. H. Eberly, "Theory of shape-preserving short pulses in inhomogeneously broadened three-leve media," Phys. Rev. A 58, R805-R808 (1998).
[CrossRef]

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of electromagnetically induced transparency," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of a self-induced-transparency soliton and a nonlinear Schrdinger soliton in an erbium-doped fiber," Phys. Rev. A 44, 5973-5987 (1991).
[CrossRef] [PubMed]

M. Nakazawa, Y. Kimura, K. Kurokawa, and K. Suzuki, "Self-induced-transparency solitons in an erbium-doped fiber waveguide," Phys. Rev. A 45, R23-R26 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967)
[CrossRef]

G. Vemuri, G. S. Agarwal, and K. V. Vasavada, "Cloning, Dragging, and Parametric Amplification of Solitons in a Coherently Driven, Nonabsorbing System," Phys. Rev. Lett. 79, 3889-3892 (1997).
[CrossRef]

R. Grobe, F. T. Hioe, and J. H. Eberly, "Formation of Shape-Preserving Pulses in a Nonlinear Adiabatically Integrable System," Phys. Rev. Lett. 73, 3183-3186 (1994).
[CrossRef] [PubMed]

M. Nakazawa, E. Yamada, and H. Kubota, "Coexistence of self-induced transparency soliton and nonlinear Schrdinger soliton," Phys. Rev. Lett. 66, 2625-2628 (1991).
[CrossRef] [PubMed]

Quantum Semiclassic. Opt.

J. H. Eberly, "Transmission of dressed fields in three-level media," Quantum Semiclassic. Opt. 7, 373-384 (1995).
[CrossRef]

Rev. Mod. Phys.

G. L. Lamb, Jr., "Analytical Descriptions of Ultrashort Optical Pulse Propagation in a Resonant Medium," Rev. Mod. Phys. 43, 99-124 (1971).
[CrossRef]

H. A. Haus, "Physical interpretation of inverse scattering formalism applied to self-induced transparency," Rev. Mod. Phys. 51, 331-339 (1971).
[CrossRef]

Other

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann, "Coherent interaction effects in pulses propagating through a doped nonlinear dispersive medium," Phys. Rev. E 65, 036617-1-036617-6 (2002).
[CrossRef]

J. H. Eberly and V. V. Kozlov, " Wave Equation for Dark Coherence in Three-Level Media," Phys. Rev. Lett. 88, 243604-1-243604-4 (2002).
[CrossRef]

G. P. Agarwal, Nonlinear Fiber Optics 2nd Ed., (Academic Press, San Diego CA 1995).

Q. Yang, J. T. Seo, B. Tabibi, and H. Wang, "Slow Light and Superluminality in Kerr Media without a Pump," Phys. Rev. Lett. 95, 063902-1-063902-4 (2005).
[CrossRef]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, "Resonant Optical Interactions with Molecules Confined in Photonic Band-Gap Fibers," Phys. Rev. Lett. 94, 093902-1-093902-4 (2005).
[CrossRef]

S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, and B. J. Kirby, "Low-Light-Level Optical Interactions with Rubidium Vapor in a Photonic Band-Gap Fiber," Phys. Rev. Lett. 97, 023603-1-023603-4 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

The schematics of three level Λ system interacting with two fields corresponding to Rabi frequencies 2G and 2g, respectively. The single photon detuning is denoted by Δ.

Fig. 2.
Fig. 2.

(a) Propagation of the sech pulses with an area 2π at different propagation distances in a single mode fibre coupled with resonant nonlinearity. The Fig. 2(b) shows the break-up of a pulse with area 4π into two pulses with area 2π. The different parameters used in the numerical simulation are as follows: group velocity dispersion β 1=-0.5 and Kerr nonlinearity γ 1=1.

Fig. 3.
Fig. 3.

Stable propagation of (a) bright and (b) dark solitons in a three level medium in presence of nonresonant nonlinearity and fiber dispersion. The different parameters are as follows: Input intensities A2=4, B2=3, pulse width σ=1, Kerr coefficient γ 1=γ 2=1, group velocity dispersion parameter β 1=1.0, β 2=-2.5, coupling constant η 1=η 2=1, single photon detuning Δσ=0. Note the opposite signs of the GVD at the two mode frequencies.

Fig. 4.
Fig. 4.

Growth of instability for (a) bright and (b) dark solitons in a three level medium in presence of nonresonant nonlinearity and fiber dispersion. Parameters are the same as in Fig. 3, except that now group velocity dispersion β 1(=-0.25) and β 2(=-1.25) have the same sign.

Fig. 5.
Fig. 5.

Spatio-temporal evolution of the (a) bright and (b) dark solitons in (i) a two-mode fiber (solid curve), (ii) 3-level system (dashed-dot curves) and (iii) in a doped fiber (dotted curves). Cases (ii) and (iii) are plotted for Δσ=0 and 5, respectively. Case (ii) also corresponds to a doped fiber with Δσ=0. The other parameters are as follows A2=4, B2=3, σ=1, η 1=η 2=1, β 1=1, β 2=-2.5

Fig. 6.
Fig. 6.

=(v -1 g -c -1) for a doped fiber as a function of normalized detuning Δσ for η 1=1 and for two values of η 2, namely, η 2=1 (solid line) and η 2=2 (dashed line). The results for a 3-level system with η 1=1 and η 2=2 is also given by the solid curve. The other parameters are as in Fig.5.

Equations (31)

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

E i ( z , t ) = i ( z , t ) e i ( ω i t k i z ) + c . c , ( i = 1 , 2 ) .
C ˙ 1 = i Δ C 1 + i G C 2 + i g C 3 ,
C ˙ 2 = i G * C 1 ,
C ˙ 3 = i g * C 1 ,
2 g = 2 d 13 · 1 h , 2 G = 2 d 12 · 2 h ,
P = [ χ ( 1 ) + χ ( 3 ) ( 1 2 + 2 2 2 ) ] 1
g z = i β 1 2 g t 2 + i γ 1 ( g 2 + 2 G 2 ) g + i η 1 C 3 * C 1
G z = i β 2 2 G t 2 + i γ 2 ( G 2 + 2 g 2 ) G + i η 2 C 2 * C 1 ,
η i η = 4 π N ω i d 2 c h _ , ( i = 1 , 2 ) .
g = A sech ( Kz t σ ) e i ( p 1 z Ω 1 t )
G = B tanh ( Kz t σ ) e i ( p 2 z Ω 2 t )
C 1 e i Ω 1 t
C 2 e i ( Ω 1 Ω 2 ) t
C 3 = α tanh ( Kz t σ ) + ( 1 α )
C 1 = ( i α A σ ) sech ( Kz t σ ) e i ( p 1 z Ω t )
C 2 = ( α B A ) sech ( Kz t σ ) e i ( p 1 p 2 ) z
i ( 1 α ) = α ( Ω Δ ) A 2 σ
A 2 B 2 = 1 σ 2
α = A 2 σ 2 A 2 σ 2 + i ( Δ Ω ) σ ,
p 1 = α 2 η 1 ( Δ Ω ) A 4 σ 2 + 2 B 2 γ 1 + β 1 ( Ω 2 1 σ 2 )
p 2 = γ 2 B 2 + β 2 Ω 2
K = η 1 α 2 σ A 2 + 2 β 1 Ω σ = η 2 α 2 σ A 2 + 2 β 2 Ω σ
2 β 1 γ 1 σ 2 + ( A 2 2 B 2 ) = 0
2 β 2 γ 2 σ 2 + ( 2 A 2 B 2 ) = 0
α 2 A 2 ( η 1 η 2 ) + 2 Ω ( β 1 β 2 ) = 0 .
Ω = α 2 A 2 ( η 2 η 1 ) 2 ( β 1 β 2 ) .
3 + 2 β 1 γ 1 + 2 β 2 γ 2 = 0 .
c v g 1 = K σ c = η 1 σ 2 ( A σ ) 2 c ( A σ ) 4 + ( Δ σ Ω σ ) 2 + 2 β 1 Ω c .
A 2 σ 2 = 1 , 1 + 2 β γ = 0 .
α = 1 , K = η σ , p = β σ 2 .
τ = t σ K z , ζ = z z 0 , z 0 = π σ 2 2 β 2 .

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