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, M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
    [CrossRef]
  2. D. N. Christodoulides, 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, 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, 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, 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, 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, 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, V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).
  13. K. Ohkuma, Y. H. Ichikawa, 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, 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, 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, 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, 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, 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, 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, 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, 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, V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

1983 (1)

D. Anderson, 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)

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

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

1972 (1)

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

Abramowitz, M.

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

Joseph, R. I.

D. N. Christodoulides, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, V. N. Serkin, “Decay of optical solitons,” JETP Lett. 42, 87–91 (1985).

Shabat, A. B.

V. E. Zakharov, 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, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 567.

Tzoar, N.

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

Vatarescu, A.

C. Pask, 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, 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, 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, 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, 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, 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, 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)

D. Anderson, 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]

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

Sov. Phys. JETP (1)

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

Other (5)

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, 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.

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

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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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