Abstract

First, we recall how two exact solutions of the nonlinear Schrödinger equation that describe (i) soliton propagation on a cw background and (ii) collision between two dark pulses can be obtained by using a common direct integration method. Second, we analyze the amplification–absorption effects on the first solution in the adiabatic approximation.

© 1993 Optical Society of America

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References

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  1. See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).
  2. K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).
  3. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  4. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).
  5. T. Kawata and H. Inoue, J. Phys. Soc. Jpn. 44, 1722 (1978).
    [CrossRef]
  6. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
    [CrossRef]
  7. H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
    [CrossRef]
  8. D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992); Phys. Rev. A 45, 6730 (1992).
    [CrossRef]
  9. N. N. Akhmediev and S. Wabnitz, J. Opt. Soc. Am. B 9, 236 (1992).
    [CrossRef]
  10. L. Gagnon and P. A. Bélanger, Phys. Rev. A 43, 6187 (1991).
    [CrossRef] [PubMed]
  11. M. Lisak, D. Anderson, and B. A. Malomed, Opt. Lett. 16, 1936 (1991).
    [CrossRef] [PubMed]
  12. W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, and R. N. Thurston, J. Opt. Soc. Am. B 6, 329 (1989).
    [CrossRef]
  13. R. N. Thurston and A. M. Weiner, J. Opt. Soc. Am. B 8, 471 (1991).
    [CrossRef]
  14. C. Paré, L. Gagnon, and P. A. Bélanger, Opt. Commun. 74, 228 (1989).
    [CrossRef]

1992 (2)

D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992); Phys. Rev. A 45, 6730 (1992).
[CrossRef]

N. N. Akhmediev and S. Wabnitz, J. Opt. Soc. Am. B 9, 236 (1992).
[CrossRef]

1991 (3)

1989 (2)

1988 (1)

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

1987 (1)

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
[CrossRef]

1985 (1)

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).

1978 (1)

T. Kawata and H. Inoue, J. Phys. Soc. Jpn. 44, 1722 (1978).
[CrossRef]

1973 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Agrawal, G. P.

See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).

Akhmediev, N. N.

N. N. Akhmediev and S. Wabnitz, J. Opt. Soc. Am. B 9, 236 (1992).
[CrossRef]

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
[CrossRef]

Anderson, D.

Bélanger, P. A.

L. Gagnon and P. A. Bélanger, Phys. Rev. A 43, 6187 (1991).
[CrossRef] [PubMed]

C. Paré, L. Gagnon, and P. A. Bélanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Blow, K. J.

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).

Doran, N. J.

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).

Eleonskii, V. M.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
[CrossRef]

Gagnon, L.

L. Gagnon and P. A. Bélanger, Phys. Rev. A 43, 6187 (1991).
[CrossRef] [PubMed]

C. Paré, L. Gagnon, and P. A. Bélanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Hadachira, H.

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

Hawkins, R. J.

Heritage, J. P.

Inoue, H.

T. Kawata and H. Inoue, J. Phys. Soc. Jpn. 44, 1722 (1978).
[CrossRef]

Kawata, T.

T. Kawata and H. Inoue, J. Phys. Soc. Jpn. 44, 1722 (1978).
[CrossRef]

Kulagin, N. E.

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
[CrossRef]

Lisak, M.

Malomed, B. A.

McLaughlin, D. W.

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

Mihalache, D.

D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992); Phys. Rev. A 45, 6730 (1992).
[CrossRef]

Moloney, J. V.

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

Newell, A. C.

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

Panoiu, N. C.

D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992); Phys. Rev. A 45, 6730 (1992).
[CrossRef]

Paré, C.

C. Paré, L. Gagnon, and P. A. Bélanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Thurston, R. N.

Tomlinson, W. J.

Wabnitz, S.

Weiner, A. M.

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

J. Math. Phys. (2)

H. Hadachira, D. W. McLaughlin, J. V. Moloney, and A. C. Newell, J. Math. Phys. 29, 63 (1988).
[CrossRef]

D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992); Phys. Rev. A 45, 6730 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. Soc. Jpn. (1)

T. Kawata and H. Inoue, J. Phys. Soc. Jpn. 44, 1722 (1978).
[CrossRef]

Opt. Commun. (1)

C. Paré, L. Gagnon, and P. A. Bélanger, Opt. Commun. 74, 228 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

K. J. Blow and N. J. Doran, Phys. Lett. 107A, 55 (1985).

Phys. Rev. A (1)

L. Gagnon and P. A. Bélanger, Phys. Rev. A 43, 6187 (1991).
[CrossRef] [PubMed]

Sov. Phys. JETP (2)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).

Teor. Mat. Fiz. (1)

N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Teor. Mat. Fiz. 72, 183 (1987) [Theor. Math. Phys. 72, 809 (1987)].
[CrossRef]

Other (1)

See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, 1989).

Cited By

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Figures (6)

Fig. 1
Fig. 1

Solution (2.16) for ν0 = 0.5 and ν = 1: (a) field amplitude; (b) principal value of Arg[Q(t, z) + iΔ(z)]; and (c) frequency spectrum at z = 0 (dashed curve) and z = T/2 (solid curve).

Fig. 2
Fig. 2

Field amplitude of Eq. (2.16) for ν0 = 1.625 and ν = 21/2.

Fig. 3
Fig. 3

Solution of Eq. (2.29) for ν0 = 0.5 and ν = 1: (a) field amplitude at z = 0 (dashed curve) and z = 3 (solid curve); (b) principal value of Arg[Q(t, z) + iΔ(z)]; and (c) frequency spectrum.

Fig. 4
Fig. 4

Field amplitude of Eq. (2.29) for ν0 = 0.11/2 and ν = 1 at z = 0 (dashed curve) and z = 3 (solid curve).

Fig. 5
Fig. 5

Evolution of ν0, η, and ν according to relations (3.2)–(3.4) for γ0 = 0.1, γ2 = 0.01, and γn = 0.

Fig. 6
Fig. 6

Evolution of ν0, η, and ν according to relations (3.2)–(3.4) for γ0 = 0.1, γ2 = 0.01, and γn = 0.002.

Equations (45)

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i q z + q t t + 2 q q 2 = 0 ,             = ± 1 ,
i q z + q t t + 2 q q 2 = i R ( q ) ,
R ( q ) = γ 0 q + γ 2 q t t - γ n q q 2 .
d d z - q 2 d t = 2 Re - q * R ( q ) d t
u ( t , z ) - a 0 ( z ) v ( t , z ) - b 0 ( z ) = 0.
a 0 = cotan φ ( z ) ,
b 0 = - Δ ( z ) [ sin φ ( z ) ] ,
u = Q ( t , z ) cos φ ( z ) - Δ ( z ) sin φ ( z ) ,
q ( t , z ) = [ Q ( t , z ) + i Δ ( z ) ] exp [ i φ ( z ) ] .
Q t 2 = - Q 4 - ( 2 Δ 2 - φ ) Q 2 + 2 Δ z Q + h ,
Q z = Δ φ - 2 Δ ( Q 2 + Δ 2 ) ,
φ z + 4 Δ 2 = W ,
h + W Δ 2 - 3 Δ 4 = - H ,
Z z 2 = - 64 Z ( Z - α 1 ) ( Z - α 2 ) ( Z - α 3 ) ,
W = 2 ( α 1 + α 2 + α 3 ) ,
H = α 1 2 + α 2 2 + α 3 2 - 2 α 1 α 2 - 2 α 2 α 3 - 2 α 1 α 3 .
Q t 2 = - ( Q - Q 1 ) ( Q - Q 2 ) ( Q - Q 3 ) ( Q - Q 4 ) - P + ( Q ) P - ( Q ) ,
P ± ( Q ) Q 2 ± 2 ( α 3 - Z ) 1 / 2 + α 3 - α 1 - α 2 + Z ± 2 sign ( Δ z ) [ ( α 1 - Z ) ( α 2 - Z ) ] 1 / 2 .
Q 1 , 4 = ( α 1 - Z ) 1 / 2 ± 2 ( α 3 - Z ) 1 / 2 ,
Q 2 = Q 3 = - ( α 1 - Z ) 1 / 2 .
Z = α 1 α 3 sin 2 { 4 [ α 3 ( α 3 - α 1 ) ] 1 / 2 ( z - z 0 ) } α 3 - α 1 cos 2 { 4 [ α 3 ( α 3 - α 1 ) ] 1 / 2 ( z - z 0 ) } ,
Q = Q 1 ( Q 2 - Q 4 ) + Q 2 ( Q 1 - Q 4 ) sinh 2 [ ( α 3 - α 1 ) 1 / 2 ( t - t 0 ) ] Q 2 - Q 4 + ( Q 1 - Q 4 ) sinh 2 [ ( α 3 - α 1 ) 1 / 2 ( t - t 0 ) ] ,
q ( t , z ) = [ 2 η η cos ( 4 η ν z ) + i ν sin ( 4 η ν z ) ν cosh ( 2 η t ) - ν 0 cos ( 4 η ν z ) - ν 0 ] exp ( 2 i ν 0 2 z ) ,
η = ( α 3 - α 1 ) 1 / 2 ,             ν = ( α 3 ) 1 / 2 ,             ν 0 = ( α 1 ) 1 / 2
T = π / 2 η ν ,
max q ( 0 , z ) = 2 ν + ν 0 ,             η 0 ,
min q ( 0 , z ) = - 2 ν + ν 0 ,             η 0 ,
q ( ± , z ) = ν 0 .
- [ q ( t , z ) 2 - q ( ± , z ) 2 ] d t = 4 η ,
- q ( t , z ) - q ( ± , z ) 2 d t = 4 η + 2 η ν 0 I cos ( 4 η ν z ) ,
I = 4 arctan [ ν + ν 0 cos ( 4 η ν z ) ν - ν 0 cos ( 4 η ν z ) ] 1 / 2 [ ν 2 - ν 0 2 cos 2 ( 4 η ν z ) ] 1 / 2 ,
q ( t ) 2 = ν 0 2 + ( 4 η 2 / ν 2 ) sech 2 ( 2 η t ) ,
q ˜ ( ω , z ) = ( 2 π [ η cos ( 4 η ν z ) + i ν sin ( 4 η ν z ) ] × sinh { ω 2 η arccos [ - ν o ν cos ( 4 η ν z ) ] } [ ν 2 - ν 0 2 cos 2 ( 4 η ν z ) ] 1 / 2 sinh ( ω π 2 η ) - ν 0 δ ( ω ) ) × exp ( 2 i ν 0 2 z ) ,
Q 1 , 2 = ( α 3 - Z ) 1 / 2 ± 2 ( α 1 - Z ) 1 / 2 ,
Q 3 = Q 4 = - ( α 3 - Z ) 1 / 2 .
Z = α 1 α 3 sinh 2 [ 4 [ α 1 ( α 3 - α 1 ) ] 1 / 2 ( z - z 0 ) ] α 3 cosh 2 [ 4 [ α 1 ( α 3 - α 1 ) ] 1 / 2 ( z - z 0 ) ] - α 1 ,
Q = Q 2 ( Q 1 - Q 3 ) + Q 1 ( Q 2 - Q 3 ) tanh 2 ( α 3 - α 1 ) 1 / 2 ( t - t 0 ) ] Q 1 - Q 3 + ( Q 2 - Q 3 ) tanh 2 [ ( α 3 - α 1 ) 1 / 2 ( t - t 0 ) ] .
q ( t , z ) = [ 2 η η cosh ( 4 ν 0 η z ) + i ν 0 sinh ( 4 ν 0 η z ) ν 0 cosh ( 2 η t ) + ν cosh ( 4 ν 0 η z ) - ν ] × exp ( 2 i ν 2 z ) ,
q ( ± , z ) = ν
q ˜ ( ω , z ) = ( 2 π [ η cosh ( 4 η ν 0 z ) + i ν 0 sinh ( 4 η ν 0 z ) ] × sin { ω 2 η arccosh [ ν ν 0 cosh ( 4 η ν 0 z ) ] } [ ν 2 cosh 2 ( 4 η ν 0 z ) - ν 0 2 ] 1 / 2 sinh ( ω π 2 η ) - ν δ ( ω ) ) × exp ( 2 i ν 2 z ) .
q cw = ν 0 ( z ) exp [ 2 i 0 z ν 0 2 ( z ) d z ] ,
d ν 0 / d z = γ 0 ν 0 - γ n ν 0 3 .
d η / d z = 2 γ 0 η - 4 η 3 [ ( γ 2 + γ n ) W + γ n ν 2 ] ν 2 - ν 0 2 cos 2 ( 4 η ν z ) - 4 γ n ν 0 2 η ,
W = ½ ν 0 ν 2 I cos ( 4 η ν z ) + [ ν 0 2 cos 2 ( 4 η ν z ) + 2 ν 2 ] ,
ν = ( η 2 + ν 0 2 ) 1 / 2

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