Abstract

We derive analytical bright and dark similaritons of the generalized coupled nonlinear Schrödinger equations with distributed coefficients. An exact balance condition between the dispersion, nonlinearity and the gain/loss has been obtained. Under this condition, we discuss the nonlinear similariton tunneling effect.

© 2010 OSA

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References

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  1. A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).
  4. C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
    [CrossRef]
  5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
    [CrossRef]
  6. V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006).
    [CrossRef]
  7. C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010).
    [CrossRef] [PubMed]
  8. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978).
    [CrossRef]
  9. V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001).
    [CrossRef]
  10. J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008).
    [CrossRef]
  11. R. C. Yang and X. L. Wu, “Spatial soliton tunneling, compression and splitting,” Opt. Express 16, 17759–17767 (2008).
    [CrossRef] [PubMed]
  12. L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
    [CrossRef]

2010 (2)

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

C. Q. Dai, Y. Y. Wang, and J. F. Zhang, “Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation,” Opt. Lett. 35, 1437–1439 (2010).
[CrossRef] [PubMed]

2008 (2)

2006 (1)

2005 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[CrossRef]

2004 (1)

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

2001 (1)

V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

1978 (1)

A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).

Belyaeva, T. L.

V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001).
[CrossRef]

Dai, C. Q.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Harvey, J. D.

V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Jia, S. T.

Kofane, T. C.

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Li, L.

J. F. Wang, L. Li, and S. T. Jia, “Nonlinear tunneling of optical similaritons in nonlinear waveguides,” J. Opt. Soc. Am. B 25, 1254–1260 (2008).
[CrossRef]

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Li, S. Q.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Li, Z. H.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Mohamadoub, A.

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Newell, A. C.

A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978).
[CrossRef]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Porsezian, K.

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

Serkin, V. N.

V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001).
[CrossRef]

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Tappet, F.

A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Tiofacka, C. G. L.

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

Wang, J. F.

Wang, Y. Y.

Wu, X. L.

Yang, R. C.

Zhang, J. F.

Zhou, G. S.

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Appl Phys. Lett. (1)

A. Hasegawa and F. Tappet, “Transimission of stationary nonlinear optical pulses in dispersive dieletric fibers.I. anomalous dispersion,” Appl Phys. Lett. 23, 142–170 (1973).
[CrossRef]

J. Math. Phys. (1)

A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126–1133 (1978).
[CrossRef]

J. Mod. Opt. (1)

C. G. L. Tiofacka, A. Mohamadoub, T. C. Kofane, and K. Porsezian, “Exact quasi-soliton solutions and soliton interaction for the inhomogeneous coupled nonlinear Schrödinger equations,” J. Mod. Opt. 57, 261–272 (2010).
[CrossRef]

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

JETP Lett. (1)

V. N. Serkin and T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573–577 (2001).
[CrossRef]

Opt. Commun. (1)

L. Li, Z. H. Li, S. Q. Li, and G. S. Zhou, “Modulation instability and soliton on cw background in inhomogeneous optical fiber media,” Opt. Commun. 234, 169–176 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Expermental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, (New York: Academic Press, 1993).

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Figures (4)

Fig. 1.
Fig. 1.

Evolutions of nonlinear tunneling of bright similariton (s 0 = 0.03) and soliton (s 0 = 0) in the (a) and (b) dispersion barrier with h 1 = 2, (c) dispersion well with h 1 = −0.5 and (d) nonlinear barrier with h 2 = 2. Other parameters: k = 2,z 01 = z 02 = 7, σ = ε = t 0 = d 0 = δ 10 = w 0 = 1, θ = η 1 = ξ 1 = 0.5, γ 0 = β 0 = −1.

Fig. 2.
Fig. 2.

The nonlinear tunneling of separating and interacting bright similariton pairs in the dispersion barrier with (a) and (b) h 1 = 2,z 01 = 7, (c) h 1 = 2,z 01 = 3 and (d) dispersion well with h 1 = −0.9, k = 0.6,z 01 = 5.2. The parameters are (a)η 1 = ξ 2 = 1.5, η 2 = ξ 1 = 1.4, θ 10 = −θ 20 = 2,ϕ 10 = ϕ 20 = 1, and (b)–(d)η 1 = η 2 = 1,ξ 1 = −ξ 2 = 1.5, θ 10 = −θ 20 = 5, ϕ 10 = ϕ 20 = 0. Other parameters are the same as that in Fig. 1.

Fig. 3.
Fig. 3.

Evolutions of nonlinear tunneling of dark similaritons in the (a) and (d) nonlinear barrier, and (b) and (c) dispersion barrier. The parameters are σ = −ε = γ 0 = β 0 = 1 with (a)η 1 = ξ 1 = 0.5,Ω = 1, (b)η 1 = ξ 2 = 1.5, η 2 = ξ 1 = 1.4,T 10 = −T 20 = 1, and (c) and (d) η 1 = −ξ 2 = 1.5, η 2 = ξ 1 = 1.4,T 10 = −T 20 = 10,z 02 = 8. Other parameters are the same as that in Fig. 1.

Fig. 4.
Fig. 4.

Bright similariton through the dispersion barrier [Eq. (17)] for the different system parameter g 0: (a) g 0 = 0.1, (b) g 0 = 0.055 and (c) g 0 = −0.1 with r = 1. Other parameters are the same as that in Fig. 1.

Equations (20)

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i Φ Z + ε Φ TT + σ Φ 2 Φ = 0 .
i Ψ 1 z 1 2 β ( z ) Ψ 1 tt + γ ( z ) ( Ψ 1 2 + Ψ 2 2 ) Ψ 1 = i g ( z ) Ψ 1 ,
i Ψ 2 z 1 2 β ( z ) Ψ 2 tt + γ ( z ) ( Ψ 2 2 + Ψ 1 2 ) Ψ 2 = i g ( z ) Ψ 2 ,
Ψ 1 ( z , t ) = ψ ( z , t ) cos θ , Ψ 2 ( z , t ) = ψ ( z , t ) sin θ e i ϑ ,
ψ ( z , t ) = ρ ( z ) e i ϕ ( z , t ) Φ ( Z , T ) ,
ρ = ρ 0 α 1 2 exp [ 0 z g ( s ) d s ] , ϕ = s 0 α 2 t 2 d 0 α t + d 0 2 α D ( z ) 2 ,
Z = α D ( z ) ( 2 ε w 0 2 ) , T = [ t ( d 0 + s 0 t 0 ) D ( z ) t 0 ] w ( z ) , w ( z ) = w 0 α ,
g ( z ) = γ β z β γ z 2 γ ( z ) β ( z ) s 0 α ( z ) β ( z ) 2 ,
{ Ψ 1 Ψ 2 } = ρ ( z ) e i ϕ ( z , t ) [ Ψ 0 + 2 2 ε σ Σ m = 1 n ( λ m + λ m * ) φ 1 , m ( λ m ) φ 2 , m * ( λ m ) A m ] { cos θ sin θ e i ϑ } ,
φ j , m + 1 ( λ m + 1 ) = ( λ m + 1 + λ m * ) φ j , m ( λ m + 1 ) B m A m ( λ m + λ m * ) φ j , m ( λ m ) ,
A m = φ 1 , m ( λ m ) 2 + φ 2 , m ( λ m ) 2 , B m = φ 1 , m ( λ m + 1 ) φ 1 , m * ( λ m ) + φ 2 , m ( λ m + 1 ) φ 2 , m * ( λ m ) ,
δ j = η j T 2 ε η j ξ j Z δ j 0 , κ j = ξ j T ε ( η j 2 ξ j 2 ) Z κ j 0 .
{ Ψ 1 Ψ 2 } = 2 ε σ η 1 ρ ( z ) sech δ 1 exp { i [ ϕ ( z , t ) + κ 1 ( Z , T ) ] } { cos θ sin θ e i ϑ } ,
Ψ 1 = 2 ε σ cos θ ρ ( z ) e i ϕ ( z , t ) G 1 F 1 ,
Ψ 2 = 2 ε σ sin θ ρ ( z ) e i ϕ ( z , t ) + ϑ G 1 F 1 ,
{ Ψ 1 Ψ 2 } = μ 2 ε σ ρ ( z ) e i [ ϕ ( z , t ) + φ ( Z , T ) ] ( 1 + G 2 F 2 ) { cos θ sin θ e i ϑ } ,
{ Ψ 1 Ψ 2 } = 2 ε σ ξ 1 i η 1 μ cos θ ρ ( z ) [ ξ 1 + i η 1 tanh ( δ 1 ) ] e i [ ϕ ( z , t ) + φ ( Z , T ) ] { cos θ sin θ e i ϑ } .
β ( z ) = 1 + h 1 sech 2 [ k ( z z 01 ) ] , γ ( z ) = γ 0 ,
β ( z ) = β 0 , γ ( z ) = 1 + h 2 sech 2 [ k ( z z 02 ) ] ,
β ( z ) = r e g 0 z + h 1 sech 2 [ k ( z z 01 ) ] , γ ( z ) = γ 0 .

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