Abstract

The propagation of light pulses as solitons in optical fibers may form the basis of a viable means of communication. We show here from the general two-soliton function that solitons in fibers exert forces on their neighbors that decrease exponentially with the distance between them and depend sinusoidally on their relative phase. These forces account for the displacements suffered by solitons during collisions, and their effects must be taken into account in system design.

© 1983 Optical Society of America

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References

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  1. V. E. Zahkarov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971); Sov. Phys. JETP 34, 62, (1972).
  2. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [Crossref]
  3. J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284, (1974).
    [Crossref]
  4. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [Crossref]
  5. V. I. Karpman and V. V. Solov’ev, Physica 3D, 487 (1981).
  6. In Ref. 4, the nlS Eq. (1) was expressed with the roles of time and distance reversed. The difference in viewpoint is whether one takes a snapshot of the pulse at a given time (the way of this Letter) or sits at a particular spot and watches the pulse go by (the way of Ref. 4). Since the pulse envelope does not change appreciably while it travels its own length in the laboratory frame, no physical difference is to be expected.
  7. In Refs. 1 and 3, the N-soliton function was given in terms of 2N algebraic equations. We have used their methods, in particular the inversion of the matrix M given in Ref. 1 to reduce the number of equations by a factor of 2, and have redefined the parameters to give them more-transparent meanings.
  8. The displacement of the jth soliton by the kth when xk (t) < xj (t) was missed in Ref. 1.
  9. If Ψ = 0 and q˙=2exp(−q), then a = υ = 0. This is the case of degenerate eigenvalues mentioned briefly in Ref. (1), and we get the same logarithmic divergence, exp(q) = const. + 2t.
  10. K. J. Blow and N. J. Doran, Electron. Lett. 19, 429 (1983).
    [Crossref]

1983 (1)

K. J. Blow and N. J. Doran, Electron. Lett. 19, 429 (1983).
[Crossref]

1981 (1)

V. I. Karpman and V. V. Solov’ev, Physica 3D, 487 (1981).

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[Crossref]

1974 (1)

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284, (1974).
[Crossref]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[Crossref]

1971 (1)

V. E. Zahkarov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971); Sov. Phys. JETP 34, 62, (1972).

Blow, K. J.

K. J. Blow and N. J. Doran, Electron. Lett. 19, 429 (1983).
[Crossref]

Doran, N. J.

K. J. Blow and N. J. Doran, Electron. Lett. 19, 429 (1983).
[Crossref]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[Crossref]

Hasegawa, A.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[Crossref]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, Physica 3D, 487 (1981).

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[Crossref]

Satsuma, J.

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284, (1974).
[Crossref]

Shabat, A. B.

V. E. Zahkarov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971); Sov. Phys. JETP 34, 62, (1972).

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, Physica 3D, 487 (1981).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[Crossref]

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[Crossref]

Yajima, N.

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284, (1974).
[Crossref]

Zahkarov, V. E.

V. E. Zahkarov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971); Sov. Phys. JETP 34, 62, (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[Crossref]

Electron. Lett. (1)

K. J. Blow and N. J. Doran, Electron. Lett. 19, 429 (1983).
[Crossref]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[Crossref]

Physica (1)

V. I. Karpman and V. V. Solov’ev, Physica 3D, 487 (1981).

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, Prog. Theor. Phys. Suppl. 55, 284, (1974).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zahkarov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971); Sov. Phys. JETP 34, 62, (1972).

Other (4)

In Ref. 4, the nlS Eq. (1) was expressed with the roles of time and distance reversed. The difference in viewpoint is whether one takes a snapshot of the pulse at a given time (the way of this Letter) or sits at a particular spot and watches the pulse go by (the way of Ref. 4). Since the pulse envelope does not change appreciably while it travels its own length in the laboratory frame, no physical difference is to be expected.

In Refs. 1 and 3, the N-soliton function was given in terms of 2N algebraic equations. We have used their methods, in particular the inversion of the matrix M given in Ref. 1 to reduce the number of equations by a factor of 2, and have redefined the parameters to give them more-transparent meanings.

The displacement of the jth soliton by the kth when xk (t) < xj (t) was missed in Ref. 1.

If Ψ = 0 and q˙=2exp(−q), then a = υ = 0. This is the case of degenerate eigenvalues mentioned briefly in Ref. (1), and we get the same logarithmic divergence, exp(q) = const. + 2t.

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Equations (19)

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i u / t = 1 2 2 u / x 2 + | u | 2 u .
u = sech ( x ) exp ( i t / 2 ) .
u = A sech [ A ( x V t ) ] exp [ i V x + i ( A 2 V 2 ) t / 2 ]
u ( x , t ) = j = 1 N u j ( x , t )
k 1 N M j k ( γ J + γ k * ) u k = 1 , j = 0 N ,
u = A j sech [ A j ( x x j ) + q j ] exp [ i ( ϕ j + Ψ j ) ] ,
q j + i Ψ j = k j ± ln [ A j + A k + i ( V j V k ) A j A k + i ( V j V k ) ] ,
u exp ( i θ ) 2 [ ( 1 + υ 2 ) ( 1 a 2 ) ] 1 / 2 = ρ exp ( x ) cosh ( z β * i ϕ ) + ρ * exp ( x ) cosh ( z + β + i ϕ ) | ρ | 2 cosh ( 2 x ) + ( 1 + υ 2 ) cosh ( 2 z ) + ( 1 a 2 ) cos ( 2 ϕ ) ,
u exp [ ( i θ + q ˙ ) ] { ( 1 Ψ ˙ ) sech [ ( 1 Ψ ˙ ) x q ] exp [ i ( q ˙ x Ψ ) ] + ( 1 + Ψ ˙ ) sech [ ( 1 + Ψ ˙ ) x + q ] exp [ i ( Ψ q ˙ x ) ] } ,
ρ exp ( q + i Ψ ) = 2 cosh ( ζ ) ,
cosh [ ζ + ρ ( x + p ) ] cosh ( ζ ) exp [ ρ ( x + p ) tanh ( ζ ) ] ,
q ¨ = 4 exp ( 2 q ) cos ( 2 Ψ ) ,
Ψ ¨ = 4 exp ( 2 q ) sin ( 2 Ψ ) ,
ρ = [ 4 exp ( 2 q i 2 Ψ ) + ( Ψ ˙ i q ˙ ) 2 ] 1 / 2 ,
b 0 + i ϕ 0 = tanh 1 [ ( Ψ ˙ i q ˙ ) / ρ ] t = 0 .
exp ( q ) = ( 2 / a ) [ cosh ( 2 b 0 ) + cos ( 2 a t ) ] 1 / 2 .
a + i υ = [ 4 exp ( 2 q 0 ) q ˙ 0 2 ] 1 / 2 ,
exp ( q ) = ( 2 / a ) | cos ( a t ) | .
2 q 0 = 2 . 3 d exp [ 1 + ln ( z / z 0 ) ]

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