Abstract

We show that the soliton self-frequency shift (SSFS) law as well as an approximate analytical description of the pulse deformation can be obtained on the basis of a Galilean-like symmetry of the simplest SSFS model.

© 1990 Optical Society of America

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References

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  1. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  2. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]
  3. M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, D. S. Chemla, Opt. Lett. 14, 370 (1989).
    [CrossRef] [PubMed]
  4. K. J. Blow, N. J. Doran, D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
    [CrossRef]
  5. L. Gagnon, P. Winternitz, J. Phys. A 22, 469 (1989).
    [CrossRef]
  6. P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
    [CrossRef]
  7. E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chaps. 2 and 14.
  8. M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 3.
    [CrossRef]
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.
  10. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.
  11. Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
    [CrossRef]
  12. L. Gagnon, “Some exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B (to be published).

1989 (2)

1988 (1)

1987 (1)

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

1986 (1)

1972 (1)

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

Ablowitz, M. J.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 3.
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.

Bar-Joseph, I.

Blow, K. J.

Chemla, D. S.

Doran, N. J.

Gagnon, L.

L. Gagnon, P. Winternitz, J. Phys. A 22, 469 (1989).
[CrossRef]

L. Gagnon, “Some exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B (to be published).

Gordon, J. P.

Hasegawa, A.

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

Ince, E. L.

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chaps. 2 and 14.

Islam, M. N.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.

Kodama, Y.

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

Olver, P. J.

P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
[CrossRef]

Segur, H.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 3.
[CrossRef]

Shabat, A. B.

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

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.

Sucha, G.

Wegener, M.

Winternitz, P.

L. Gagnon, P. Winternitz, J. Phys. A 22, 469 (1989).
[CrossRef]

Wood, D.

Zakharov, V. E.

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

IEEE J. Quantum Electron. (1)

Y. Kodama, A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

L. Gagnon, P. Winternitz, J. Phys. A 22, 469 (1989).
[CrossRef]

Opt. Lett. (2)

Sov. Phys. JETP (1)

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

Other (6)

L. Gagnon, “Some exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B (to be published).

P. J. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, Berlin, 1986).
[CrossRef]

E. L. Ince, Ordinary Differential Equations (Dover, New York, 1956), Chaps. 2 and 14.

M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 3.
[CrossRef]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 10.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Sec. 7.11.

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Figures (1)

Fig. 1
Fig. 1

Normalized intensity distribution τ02|u|2 of solution (14) for τd/τ0 = 0 (dashed curve) and τd/τ0 = 0.3 (solid curve).

Equations (18)

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i u z + 1 2 u tt + u | u | 2 t d u ( | u | 2 ) t = 0 ,
z = λ + z , t = b 2 λ 2 + bzλ + t , u ( z , t ) = u ( z , t ) × exp [ i ( b 2 6 λ 3 + b 2 2 λ 2 z + btλ ) ] ,
[ z ] + a [ i ( u u u * u * ) ] + b [ z t + i ( u u u * u * ) ] .
t = t + ν 0 z , z = z , u ( z , t ) = u ( z , t ) exp [ i ( ν 0 t + ν 0 2 2 z ) ] ,
ξ = t b 2 z 2 ,
f = u ( z , t ) exp [ i ( b 2 3 z 3 bzt + az ) ] ,
af bξf + 1 2 f ξξ + f | f | 2 t d f ( | f | 2 ) ξ = 0 .
χ = S 0 M 2 d ξ ,
( a ) M + 1 2 M ξξ S 0 2 M 8 + M 3 2 t d M 2 M ξ = 0 ,
M ( 0 ) = 2 a sech ( 2 a ξ ) , a < 0 ,
M = M ( 0 ) + t d M ( 1 ) = M ( 0 ) + t d F ( x ) sech ( x ) , x = 2 a ξ ,
t d [ F xx 2 F x tanh ( x ) + 4 F sech 2 ( x ) ] = S 0 2 a ( 2 a ) 3 / 2 1 sech 4 ( x ) b a x + 8 t d a sech 2 ( x ) tanh ( x ) .
F = Ax + { B x 2 + C ln [ sech ( x ) ] } tanh ( x ) ,
A = 16 15 a , B = 8 15 a , C = 8 5 a , S 0 = 0 .
b = 32 15 t d a 2 8 15 t d τ 0 2 ,
u = 1 τ 0 { 1 + 8 t d 15 τ 0 2 ξ [ 4 t d 15 τ 0 3 ξ 2 12 t d 15 τ 0 ln sech ( ξ τ 0 ) ] × tanh ( ξ τ 0 ) } sech ( ξ τ 0 ) exp { i [ 8 t d 15 τ 0 4 zt 1 3 ( 8 t d 15 τ 0 4 ) 2 z 3 + z 2 τ 0 2 ] } ,
u = 1 τ ( z ) sech [ t v ( z ) τ ( z ) ] × exp ( i { ω ( z ) [ t υ ( z ) ] α ( z ) } )
d τ d z = 0 , d ω d z = 8 t d 15 τ 4 , d α d z = 1 2 ω 2 + 1 2 τ 2 , d υ d z = ω .

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