Abstract

A spectral method that obtains the soliton and periodic solutions to the nonlinear wave equation is presented. The results show that the nonlinear group velocity is a function of the frequency shift as well as of the soliton power. When the frequency shift is a function of time, a solution in terms of the Jacobian elliptic function is obtained. This solution is periodic in nature, and, to generate such an optical pulse train, one must simultaneously amplitude- and frequency-modulate the optical carrier. Finally, we extend the method to include the effect of self-steepening.

© 1993 Optical Society of America

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References

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  1. N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
    [Crossref]
  2. D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers–beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76–78 (1985).
    [Crossref]
  3. G. P. Agrawal, Non-linear Fiber Optics (Academic, San Diego, 1989), p. 136.
  4. P. P. Banerjee, M. R. Chatterjee, and M. Maghraoui, “Spectral approach to optical propagation across a linear–nonlinear interface,” J. Opt. Soc. Am. B 7, 21–29 (1990).
    [Crossref]
  5. M. Maghraoui and P. P. Banerjee, “An exact solution to the spatial evolution of a carrier and a pair of sidebands in a cubically non-linear medium,” Opt. Commun. 83, 358–366 (1991).
    [Crossref]
  6. C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. 3, 1018–1024 (1986).
    [Crossref]
  7. P. L. Frangois, “Nonlinear propagation of ultrashort pulses in optical fibers: total field formulation in the frequency domain,” J. Opt. Soc. Am. B 8, 276–293 (1991).
    [Crossref]
  8. Y. Kodama and J. J. Ablowitz, “Perturbations of solitons and solitary waves,” Stud. Appl. Math. 64, 225–245 (1981).
  9. Y. Kodama, “Optical solitons in a monomode fiber,” J. Stat. Phys. 39, 597–620 (1985).
    [Crossref]
  10. A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1989), p. 32.
  11. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).
  12. E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).
  13. K. Ohkuma, Y. H. Ichikawa, and Y. Abe, “Soliton propagation along optical fibers,” Opt. Lett. 12, 516–518 (1987).
    [Crossref] [PubMed]
  14. E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).
  15. D. Anderson and M. Lisak, “Non-linear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
    [Crossref]
  16. H. T. Davis, Introduction to Non-linear Differential Equations and Integral Equations (Dover, New York, 1962), p. 145.
  17. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 567.
  18. G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1981), p. 114.

1991 (2)

M. Maghraoui and P. P. Banerjee, “An exact solution to the spatial evolution of a carrier and a pair of sidebands in a cubically non-linear medium,” Opt. Commun. 83, 358–366 (1991).
[Crossref]

P. L. Frangois, “Nonlinear propagation of ultrashort pulses in optical fibers: total field formulation in the frequency domain,” J. Opt. Soc. Am. B 8, 276–293 (1991).
[Crossref]

1990 (1)

1987 (2)

K. Ohkuma, Y. H. Ichikawa, and Y. Abe, “Soliton propagation along optical fibers,” Opt. Lett. 12, 516–518 (1987).
[Crossref] [PubMed]

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

1986 (1)

C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. 3, 1018–1024 (1986).
[Crossref]

1985 (3)

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers–beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76–78 (1985).
[Crossref]

Y. Kodama, “Optical solitons in a monomode fiber,” J. Stat. Phys. 39, 597–620 (1985).
[Crossref]

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

1983 (1)

D. Anderson and M. Lisak, “Non-linear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[Crossref]

1981 (2)

Y. Kodama and J. J. Ablowitz, “Perturbations of solitons and solitary waves,” Stud. Appl. Math. 64, 225–245 (1981).

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[Crossref]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Abe, Y.

Ablowitz, J. J.

Y. Kodama and J. J. Ablowitz, “Perturbations of solitons and solitary waves,” Stud. Appl. Math. 64, 225–245 (1981).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 567.

Agrawal, G. P.

G. P. Agrawal, Non-linear Fiber Optics (Academic, San Diego, 1989), p. 136.

Anderson, D.

D. Anderson and M. Lisak, “Non-linear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[Crossref]

Banerjee, P. P.

M. Maghraoui and P. P. Banerjee, “An exact solution to the spatial evolution of a carrier and a pair of sidebands in a cubically non-linear medium,” Opt. Commun. 83, 358–366 (1991).
[Crossref]

P. P. Banerjee, M. R. Chatterjee, and M. Maghraoui, “Spectral approach to optical propagation across a linear–nonlinear interface,” J. Opt. Soc. Am. B 7, 21–29 (1990).
[Crossref]

Chatterjee, M. R.

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers–beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76–78 (1985).
[Crossref]

Davis, H. T.

H. T. Davis, Introduction to Non-linear Differential Equations and Integral Equations (Dover, New York, 1962), p. 145.

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

Frangois, P. L.

Golovchenko, E. A.

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1989), p. 32.

Ichikawa, Y. H.

Jain, M.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[Crossref]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers–beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76–78 (1985).
[Crossref]

Kodama, Y.

Y. Kodama, “Optical solitons in a monomode fiber,” J. Stat. Phys. 39, 597–620 (1985).
[Crossref]

Y. Kodama and J. J. Ablowitz, “Perturbations of solitons and solitary waves,” Stud. Appl. Math. 64, 225–245 (1981).

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1981), p. 114.

Lisak, M.

D. Anderson and M. Lisak, “Non-linear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[Crossref]

Maghraoui, M.

M. Maghraoui and P. P. Banerjee, “An exact solution to the spatial evolution of a carrier and a pair of sidebands in a cubically non-linear medium,” Opt. Commun. 83, 358–366 (1991).
[Crossref]

P. P. Banerjee, M. R. Chatterjee, and M. Maghraoui, “Spectral approach to optical propagation across a linear–nonlinear interface,” J. Opt. Soc. Am. B 7, 21–29 (1990).
[Crossref]

Ohkuma, K.

Pask, C.

C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. 3, 1018–1024 (1986).
[Crossref]

Pilpetskii, A. N.

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

Prokhorov, A. M.

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

Serkin, V. N.

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 567.

Tzoar, N.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[Crossref]

Vatarescu, A.

C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. 3, 1018–1024 (1986).
[Crossref]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers–beyond the slowly varying envelope approximation,” Appl. Phys. Lett. 47, 76–78 (1985).
[Crossref]

J. Opt. Soc. Am. (1)

C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. 3, 1018–1024 (1986).
[Crossref]

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

J. Stat. Phys. (1)

Y. Kodama, “Optical solitons in a monomode fiber,” J. Stat. Phys. 39, 597–620 (1985).
[Crossref]

JETP Lett. (2)

E. A. Golovchenko, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

E. A. Golovchenko, E. M. Dianov, A. N. Pilpetskii, A. M. Prokhorov, and V. N. Serkin, “Self-effect and maximum contraction of optical femtosecond wave packets in a non-linear dispersive medium,” JETP Lett. 45, 91–95 (1987).

Opt. Commun. (1)

M. Maghraoui and P. P. Banerjee, “An exact solution to the spatial evolution of a carrier and a pair of sidebands in a cubically non-linear medium,” Opt. Commun. 83, 358–366 (1991).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (2)

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[Crossref]

D. Anderson and M. Lisak, “Non-linear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[Crossref]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in non-linear media,” Sov. Phys. JETP 34, 62–69 (1972).

Stud. Appl. Math. (1)

Y. Kodama and J. J. Ablowitz, “Perturbations of solitons and solitary waves,” Stud. Appl. Math. 64, 225–245 (1981).

Other (5)

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1989), p. 32.

G. P. Agrawal, Non-linear Fiber Optics (Academic, San Diego, 1989), p. 136.

H. T. Davis, Introduction to Non-linear Differential Equations and Integral Equations (Dover, New York, 1962), p. 145.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 567.

G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1981), p. 114.

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Figures (2)

Fig. 1
Fig. 1

Group-velocity shift as a function of the peak electric field.

Fig. 2
Fig. 2

a, Electric-field envelope as a function of the normalized quantity μ. b, Frequency shift as a function of the normalized quantity μ.

Equations (97)

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2 ψ x 2 = 1 c 2 2 ψ t 2 + 1 c 2 2 t 2 [ - χ 1 ( t - t ) ψ ( t ) d t ] + 2 n 2 n 0 c 2 2 ( ψ 2 ψ ) t 2 ,
d 2 ψ ˜ d x 2 + β 2 ( ω ) ψ ˜ + 2 n 2 n 0 ω 2 c 2 [ ψ ˜ * F ( ψ * ) * ψ ˜ ] = 0 ,
β ( ω ) = ω n ( ω ) / c
1 + χ ( 1 ) ( ω ) = n 2 ( ω ) .
β ( ω ) = β 0 + β 1 Ω + β 2 Ω 2 2 + ,
β n = ( d n β d ω n ) ω = ω 0 ,             n = 0 , 1 , ,             Ω = ω - ω 0 .
d 2 ψ ˜ d x 2 + [ β 0 2 + 2 β 0 β 1 Ω + ( β 0 β 2 + β 1 2 ) Ω 2 ] ψ ˜ + 2 n 2 n 0 ω 2 c 2 [ ψ ˜ * F ( ψ * ) * ψ ˜ ] = 0.
2 ψ x 2 + [ β 0 2 - 2 β 0 β 1 ω 0 + ( β 0 β 2 + β 1 2 ) ω 0 2 ] ψ + [ 2 β 0 β 1 - 2 ( β 0 β 2 + β 1 2 ) ω 0 ] i ψ t - ( β 0 β 2 + β 1 2 ) 2 ψ t 2 - 2 n 2 n 0 c 2 2 ( ψ 2 ψ ) t 2 = 0.
ψ ( x , t ) = - a ( ω , x ) exp [ - i ( ω t - k ( ω ) x ) ] d ω ,
k = β ˜ 0 + β ˜ 1 Ω ,
ψ ( x , t ) = exp [ - i ( ω 0 t - β ˜ 0 x ) ] - A ( Ω , x ) exp [ - i Ω ( t - β ˜ 1 x ) ] d Ω ,
d 2 A ( Ω , x ) d x 2 + 2 i ( β ˜ 0 + β ˜ 1 Ω ) d A ( Ω , x ) d x - ( β ˜ 0 + β ˜ 1 Ω ) 2 A ( Ω , x ) + [ β 0 2 - 2 β 0 β 1 ω 0 + ( β 0 β 2 + β 1 2 ) ω 0 2 ] A ( Ω , x ) + [ 2 β 0 β 1 - 2 ( β 0 β 2 + β 1 2 ) ω 0 ] ω A ( Ω , x ) + ( β 0 β 2 + β 1 2 ) ω 2 A ( Ω , x ) = - 2 n 2 n 0 ω 2 c 2 - - A ( Ω 1 , x ) A ( Ω 2 - Ω 1 , x ) A * ( Ω 2 - Ω , x ) × d Ω 1 d Ω 2 .
[ ( β 0 2 - β ˜ 0 2 ) + 2 ( β 0 β 1 - β ˜ 0 β ˜ 1 ) Ω + ( β 0 β 2 + β 1 2 - β ˜ 1 2 ) Ω 2 ] A ( Ω ) = - ( 2 n 2 n 0 ) ( ω 0 + Ω ) 2 c 2 × - - A ( Ω 1 ) A ( Ω 2 - Ω 1 ) A * ( Ω 2 - Ω ) d Ω 1 d Ω 2 .
( η 2 - 2 α Ω - δ Ω 2 ) A ( Ω ) = 2 n 2 n 0 - - A ( Ω 1 ) A ( Ω 2 - Ω 1 ) A * ( Ω 2 - Ω ) d Ω 1 d Ω 2 ,
η 2 = β ˜ 0 2 / β 0 2 - 1 ,
α = ( β 0 β 1 - β ˜ 0 β ˜ 1 ) / β 0 2 ,
δ = ( β 0 β 2 + β 1 2 - β ˜ 1 2 ) / β 0 2 ,
η 2 A ˜ + 2 i α d A ˜ d τ + δ d 2 A ˜ d τ 2 = 2 n 2 n 0 A ˜ 2 A ˜ ,
A ˜ ( τ ) = ψ ( x , t ) exp [ i ( ω 0 t - β ˜ 0 x ) ] ,
τ = - ( t - β ˜ 1 x ) .
A ˜ = ρ exp ( i θ ) ,
η 2 ρ + 2 i α ( ρ τ + i θ τ ρ ) + δ ( ρ τ τ + 2 i ρ τ θ τ + i ρ θ τ τ - ρ θ τ 2 ) = 2 ( n 2 / n 0 ) ρ 3 ,
η 2 ρ - 2 α θ τ ρ + δ ρ τ τ - δ ρ θ τ 2 = 2 ( n 2 / n 0 ) ρ 3 ,
2 α ρ τ + 2 ρ τ θ τ δ + ρ θ τ τ δ = 0.
( 2 α + 2 θ τ δ ) ρ τ = 0 ,
θ τ = - α / δ ,
θ = - α τ / δ + ν 0 ,
( η 2 + α 2 / δ ) ρ + δ ρ τ τ = 2 n 2 ρ 3 / n 0 .
ρ = [ n 0 ( η 2 + α 2 / δ ) n 2 ] 1 / 2 sech [ ( η 2 + α 2 / δ - δ ) 1 / 2 τ ] .
ψ ( x , t ) = [ n 0 ( η 2 + α 2 / δ ) n 2 ] 1 / 2 sech [ ( η 2 + α 2 / δ - δ ) 1 / 2 ( t - β ˜ 1 x ) ] × exp [ i ( α δ ) ( t - β ˜ 1 x ) + i ν 0 ] exp [ - i ( ω 0 t - β ˜ 0 x ) ] .
ρ 0 = [ ( η 2 + α 2 / δ ) ( n 0 / n 2 ) ] 1 / 2 ,
r = - α / δ .
β ˜ 0 = β 0 ( 1 + η 2 ) 1 / 2 .
r = β 0 [ ( 1 + η 2 ) 1 / 2 β ˜ 1 - β 1 ] / ( β 0 β 2 + β 1 2 - β ˜ 1 2 ) .
β ˜ 1 = [ β 0 2 ( 1 + η 2 ) + 4 r β 0 β 1 + 4 r 2 ( β 0 β 2 + β 1 2 ) ] 1 / 2 - β 0 ( 1 + η 2 ) 1 / 2 2 r .
β ˜ 1 = β 1 ( 1 + ρ 0 2 n 2 / n 0 ) 1 / 2
v ˜ g = v g ( 1 + ρ 0 2 n 2 / 2 n 0 ) ,
α ( β 1 - β ˜ 1 ) / β 0 ,
δ β 2 / β 0 ,
β ˜ 0 β 0 ( 1 + η 2 / 2 ) ,
β ˜ 1 β 1 + β 2 r ,
ψ ( x , t ) ( n 0 / n 2 ) 1 / 2 η sech { ( η / - δ ) [ t - ( β 1 + β 2 r ) x ] } × exp { - i r [ t - ( β 1 + β 2 r ) x ] + i ν 0 } × exp [ - i ( ω 0 t - β ˜ 0 x ) ] .
ψ ( x , t ) = [ n 0 ( η 2 + α 2 / δ ) 2 n 2 ] 1 / 2 tanh [ ( η 2 + α 2 / δ 2 δ ) 1 / 2 ( t - β ˜ 1 x ) ] × exp [ i ( α δ ) ( t - β ˜ 1 x ) + i ν 0 ] exp [ - i ( ω 0 t - β ˜ 0 x ) ] .
( 2 ϕ x 2 - β 1 2 2 ϕ t 2 ) + 2 β 0 [ i ( ϕ x + β 1 ϕ t ) - β 2 2 2 ϕ t 2 ] - 2 n 2 n 0 c 2 2 t 2 { ϕ 2 ϕ exp [ - i ( ω 0 t - β 0 x ) ] } = 0.
Z = 2 β 0 x ,
T = ( - β 0 / β 2 ) 1 / 2 ( t - β 1 x ) ,
q = ( n 2 / n 0 2 ) 1 / 2 ϕ ,
i q Z + 1 2 2 q T 2 + q 2 q = ( k 1 q Z T - 2 q t t ) ,
k 1 = β 1 ( - β 0 β 2 ) - 1 / 2 .
q ( T , Z ) = ξ sech [ ξ ( T + κ Z - Θ 0 ) ] × exp [ - i κ T + ( i / 2 ) ( ξ 2 - κ 2 ) Z - i σ 0 ] ,
q ( T , Z ) = q ^ ( Θ , Z 1 ; ) exp [ - i κ ( Θ - Θ 0 ) + i ( σ - σ 0 ) ] ,
Z 1 = Z
Θ T = 1 ,             Θ Z = κ , σ T = 0 ,             σ Z = 1 2 ( ξ 2 + κ 2 ) ,
1 2 2 q ^ Θ 2 + q ^ 2 q ^ - 1 2 ξ 2 q ^ = F ( q ^ ) ,
F ( q ^ ) = - i q ^ Z 1 - { Z 1 [ ( Θ - Θ 0 ) κ + σ 0 ] - 1 2 ( ξ 2 - κ 2 ) k 1 κ } q ^ + i 2 k 1 ( ξ 2 - 3 κ 2 ) q ^ Θ + k 1 κ q ^ Θ Θ .
q ^ ( Θ , Z 1 ; ) = q ^ ( 0 ) ( Θ , Z 1 ) + q ^ ( 1 ) ( Θ , Z 1 ) + ,
q ^ ( 0 ) ( Θ , Z 1 ) = ξ sech [ ξ ( Θ - Θ 0 ) ] .
d ξ d Z 1 = 0 ,             d κ d Z 1 = 0.
q ^ ( 1 ) = Φ + i Ψ ,
Φ = 1 ξ { s 0 - ( s 0 + k 1 κ ξ 2 ) tanh [ ξ ( Θ - Θ 0 ) ] } × sech [ ξ ( Θ - Θ 0 ) ] ,
Ψ = [ k 1 ( ξ 2 - 3 κ 2 ) 2 + d Θ 0 d Z 1 ] ξ ( Θ - Θ 0 ) × sech [ ξ ( Θ - Θ 0 ) ] ,
s 0 = d d Z 1 ( κ Θ 0 - σ 0 ) + 1 2 ( ξ 2 - κ 2 ) k 1 κ .
ξ = ξ 0 ,             κ = Θ 0 = σ 0 = 0
Θ 0 = - k 1 ξ 0 2 Z 1 / 2 ,             σ 0 = 0.
ψ ( x , t ) = ξ 0 ( n 0 / n 2 ) 1 / 2 sech [ ξ 0 ( - β 0 / β 2 ) 1 / 2 ( t - β 1 x ) ] × exp [ - i ( ω 0 t - β 0 x ) ] + 2 ( n 0 / n 2 ) 1 / 2 ( Φ + i Ψ ) + O ( 3 ) ,
β 0 = β 0 ( 1 + ξ 0 2 2 / 2 ) ,             β 1 = β 1 ( 1 - ξ 0 2 2 / 2 ) .
2 ( α + θ τ δ ) ρ τ = - ρ δ θ τ τ .
ρ = ρ 0 v 0 1 / 2 / ( α + θ τ δ ) 1 / 2 ,
θ τ = ( 1 / δ ) [ v 0 ( ρ 0 / ρ ) 2 - α ]
( η 2 + α 2 δ ) ρ + δ ρ τ τ - ( v 0 2 ρ 0 2 δ ) ρ - 3 = 2 n 2 ρ 3 n 0 .
ρ = ( ρ 0 2 - ( ρ 0 2 - s 2 ) sn 2 { [ - ( ρ 0 2 - s 1 ) n 2 δ n 0 ] 1 / 2 τ , l } ) 1 / 2 ,
η 2 A ˜ + 2 i α d A ˜ d τ + δ d 2 A ˜ d τ 2 = 2 n 2 n 0 A ˜ 2 A ˜ - i γ d ( A ˜ 2 A ˜ ) d τ ,
γ = 4 n 2 / n 0 ω 0 .
η 2 ρ - 2 α θ τ ρ + δ ρ τ τ - δ ρ θ τ 2 = ( 2 n 2 / n 0 ) ρ 3 + ρ 3 γ θ τ ,
2 α ρ τ + 2 ρ τ θ τ δ + ρ θ τ τ δ = - 3 ρ 2 γ ρ τ ,
( 2 α + 2 θ τ δ + 3 ρ 2 γ ) ρ τ = 0 ,
θ τ = - ( 3 ρ 2 γ + 2 α ) / 2 δ .
1 δ ( η 2 + α 2 δ ) ρ - 1 δ ( 2 n 2 n 0 - γ α δ ) ρ 3 - ( 3 γ 2 4 δ 2 ) ρ 5 + ρ τ τ = 0.
ρ = D ρ 0 ( 2 D - B ρ 0 2 ) 1 / 2 [ cosh 2 ( D τ ) + B ρ 0 2 - D 2 D - B ρ 0 2 ] - 1 / 2 ,
δ ρ τ 2 = n 2 ρ 4 n 0 - ( η 2 + α 2 δ ) ρ 2 - ( v 0 2 ρ 0 4 δ ) ρ - 2 + K ,
( ρ ρ τ ) 2 = 1 δ [ n 2 ρ 6 n 0 - ( η 2 + α 2 δ ) ρ 4 - v 0 2 ρ 0 4 δ + K ρ 2 ] .
K = ( η 2 + α 2 + v 0 2 δ ) ρ 0 2 - n 2 ρ 0 4 n 0 .
u τ 2 = 4 δ [ n 2 u 3 n 0 - ( η 2 + α 2 δ ) u 2 - v 0 2 u 0 2 δ + K u ] .
u τ 2 = ( 4 n 2 / δ n 0 ) ( u - u 0 ) ( u - s 1 ) ( u - s 2 ) ,
s 1 , 2 = 1 2 b 1 2 ( b 2 - 4 v 0 2 u 0 n 0 n 2 δ ) 1 / 2 ,
b = ( η 2 + α 2 / δ ) ( n 0 / n 2 ) - ρ 0 2 .
g τ 2 = ( - 4 n 2 / δ n 0 ) g ( u 0 - s 1 - g ) ( u 0 - s 2 - g ) .
h τ 2 = [ - n 2 ( u 0 - s 1 ) / n 0 δ ] ( 1 - h 2 ) ( 1 - l 2 h 2 ) ,
l = ( u 0 - s 2 ) / ( u 0 - s 1 ) .
h = sn { [ - n 2 ( u 0 - s 1 ) n 0 δ ] 1 / 2 τ , l } ,
u = ( u 0 - ( u 0 - s 2 ) sn 2 { [ - ( u 0 - s 1 ) n 2 δ n 0 ] 1 / 2 τ , l } ) 1 / 2 .
- D ρ 2 + B ρ 4 - C ρ 6 + ρ τ 2 + C 1 = 0 ,
D = - 1 δ ( η 2 + α 2 δ ) ,
B = - 1 2 δ ( 2 n 2 n 0 - γ α δ ) ,
C = γ 2 4 δ 2 ,
D = B ρ 0 2 - C ρ 0 4 .
ρ 0 ρ d ρ ρ [ C ( ρ 4 - ρ 0 4 ) - B ( ρ 2 - ρ 0 2 ) ] 1 / 2 = ± τ .

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