Abstract

We obtain exact self-similar solutions to an inhomogeneous nonlinear Schrödinger equation, describing propagation of optical pulses in fiber amplifiers with distributed dispersion and gain. We show that there exists a one-to-one correspondence between such self-similar waves and solitons of the standard, homogeneous, nonlinear Schrödinger equation if a certain compatibility condition is satisfied. As this correspondence guarantees the stability of the novel self-similar waves, we refer to them as similaritons. We demonstrate that, the character of similariton interactions crucially depends on the sign of the similariton phase chirp. In particular, we show that the similariton interactions can under certain conditions lead to the formation of molecule-like bound states of two similaritons.

© 2007 Optical Society of America

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  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University Press, Cambridge, UK, 2003).
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, CA, 2003).
  3. L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, San Diego, CA, 2006).
  4. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, San Diego, CA, 2007).
  5. H.-H. Chen and C.-S. Liu, ‘Solitons in nonuniform nedia," Phys. Rev. Lett. 37, 693-697 (1976).
    [CrossRef]
  6. R. Balakrishnan, "Soliton propagation in nonuniform media," Phys. Rev. A 32, 1144-1149 (1985).
    [CrossRef] [PubMed]
  7. V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion," IEEE J. Sel. Top. Quantum Electron 8, 418-431 (2002).
    [CrossRef]
  8. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
    [CrossRef] [PubMed]
  9. G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics (Cambridge University Press, Cambridge, UK, 1996).
  10. J. D. Moores, "Nonlinear compression of chirped solitary waves with and without phase modulation, Opt. Lett. 21, 555-557 (1996).
    [CrossRef] [PubMed]
  11. V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502-4505 (2000).
    [CrossRef] [PubMed]
  12. V. I. Kruglov, A. C. Peacock, J. M. Dudley, J. D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25, 1753-1755 (2000).
    [CrossRef]
  13. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schr¨odinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  14. C. Finot and G. Millot, "Collisions between similaritons in optical fiber amplifiers," Opt. Express 13, 7653-7665 (2005).
    [CrossRef] [PubMed]
  15. V. I. Kruglov and J. D. Harvey, "Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters," J. Opt. Soc. Am. B 23, 2541-2550 (2006).
    [CrossRef]
  16. S. A. Ponomarenko and G. P. Agrawal, "Do solitonlike self-similar waves exist in nonlinear optical media?" Phys. Rev. Lett. 97, 013901 (2006).
    [CrossRef] [PubMed]
  17. K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
    [CrossRef] [PubMed]
  18. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
  19. B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
    [CrossRef]
  20. G. P. Agrawal, Lightwave technology: Telecommunication Systems (Wiley, Hoboken, NJ, 2005).
    [CrossRef]

2007 (1)

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

2006 (2)

2005 (2)

C. Finot and G. Millot, "Collisions between similaritons in optical fiber amplifiers," Opt. Express 13, 7653-7665 (2005).
[CrossRef] [PubMed]

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

2003 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schr¨odinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2002 (2)

V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion," IEEE J. Sel. Top. Quantum Electron 8, 418-431 (2002).
[CrossRef]

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

2000 (2)

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502-4505 (2000).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, J. D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

1996 (1)

1985 (1)

R. Balakrishnan, "Soliton propagation in nonuniform media," Phys. Rev. A 32, 1144-1149 (1985).
[CrossRef] [PubMed]

1976 (1)

H.-H. Chen and C.-S. Liu, ‘Solitons in nonuniform nedia," Phys. Rev. Lett. 37, 693-697 (1976).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, "Do solitonlike self-similar waves exist in nonlinear optical media?" Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Balakrishnan, R.

R. Balakrishnan, "Soliton propagation in nonuniform media," Phys. Rev. A 32, 1144-1149 (1985).
[CrossRef] [PubMed]

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

Chen, H.-H.

H.-H. Chen and C.-S. Liu, ‘Solitons in nonuniform nedia," Phys. Rev. Lett. 37, 693-697 (1976).
[CrossRef]

Dudley, J. M.

Finot, C.

Gao, Y. T.

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

Harvey, J. D.

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion," IEEE J. Sel. Top. Quantum Electron 8, 418-431 (2002).
[CrossRef]

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502-4505 (2000).
[CrossRef] [PubMed]

Hulet, R. G.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Kruglov, V. I.

Liu, C.-S.

H.-H. Chen and C.-S. Liu, ‘Solitons in nonuniform nedia," Phys. Rev. Lett. 37, 693-697 (1976).
[CrossRef]

Millot, G.

Moores, J. D.

Partridge, G. B.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schr¨odinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, J. D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, "Do solitonlike self-similar waves exist in nonlinear optical media?" Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion," IEEE J. Sel. Top. Quantum Electron 8, 418-431 (2002).
[CrossRef]

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502-4505 (2000).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Shan, W. R.

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

Strecker, K. E.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Tian, B.

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

Truscott, A. G.

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Wei, G. M.

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Zhang, C. Y.

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

Eur. Phys. J. B (1)

B. Tian, W. R. Shan, C. Y. Zhang, G. M. Wei, and Y. T. Gao, "Transformations for a generalized variablecoefficient nonlinear Schrodinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation," Eur. Phys. J. B 47, 329-332 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron (1)

V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion," IEEE J. Sel. Top. Quantum Electron 8, 418-431 (2002).
[CrossRef]

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

Nature (1)

K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, "Formation and propagation of matter-wave soliton trains," Nature 417, 150-153 (2002).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

R. Balakrishnan, "Soliton propagation in nonuniform media," Phys. Rev. A 32, 1144-1149 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

H.-H. Chen and C.-S. Liu, ‘Solitons in nonuniform nedia," Phys. Rev. Lett. 37, 693-697 (1976).
[CrossRef]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous Solitons in External Potentials," Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schr¨odinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502-4505 (2000).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, "Do solitonlike self-similar waves exist in nonlinear optical media?" Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Other (6)

G. I. Barenblatt, Scaling, self-similarity, and intermediate asymptotics (Cambridge University Press, Cambridge, UK, 1996).

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering (Cambridge University Press, Cambridge, UK, 2003).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, CA, 2003).

L. F. Mollenauer and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press, San Diego, CA, 2006).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, San Diego, CA, 2007).

G. P. Agrawal, Lightwave technology: Telecommunication Systems (Wiley, Hoboken, NJ, 2005).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Evolution of the fundamental (a) bright and (b) dark similaritons inside a fiber with constant dispersion. The sign of initial chirp of |c 0|=0.1 ensures c 0β >0. The initial chirp is reversed in parts (c) and (d) so that c 0β < 0 and the fiber exhibits loss instead of the gain.

Fig. 2.
Fig. 2.

(a). Amplifier gain g(z) and (b) similariton width w(z) as a function of z/L D for several values of σ in the case c 0 = -0.1 and β2 < 0.

Fig. 3.
Fig. 3.

Evolution of the fundamental (a) bright and (b) dark similaritons over 8LD in the specific case of σ = 0.1.

Fig. 4.
Fig. 4.

Collision of two bright similaritons inside a constant-dispersion fiber (top) and a dispersion-decreasing fiber (bottom) with π = 0.2. In both cases c 0 = -0.1.

Fig. 5.
Fig. 5.

Collision of two bright similaritons inside a constant dispersion fiber. The similariton parameters are μ = 1, ν = 1, and c 0 = 0.

Fig. 6.
Fig. 6.

Formation of a two-similariton bound state upon collision of two bright similaritons inside a dispersion-decreasing fiber with μ = 3, ν = 1, π = 0.8, and c 0 = 0.

Fig. 7.
Fig. 7.

Formation of a two-similariton bound state upon collision of two bright similaritons inside a dispersion decreasing fiber with c 0 = 0.2. All other parameters are identical to those used in Fig.6.

Equations (23)

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i U z i g ( z ) 2 U β ( z ) 2 2 U τ 2 + γ ( z ) U 2 U = 0 ,
U ( τ , z ) = A ( z ) Ψ [ τ τ c ( z ) w ( z ) , ζ ( z ) ] exp [ i Φ ( τ , z ) ] ,
Φ ( τ , z ) = c ( z ) τ 2 2 + b ( z ) τ + d ( z ) ,
i ζ Ψ 1 2 χχ 2 Ψ + Ψ 2 Ψ = 0 ,
χ ( τ , z ) = [ τ τ c ( z ) ] w ( z ) .
ζ ( z ) = D ( z ) w 0 2 [ 1 c 0 D ( z ) ] , w ( z ) = w 0 [ 1 c 0 D ( z ) ] ,
A ( z ) = w ( z ) 1 [ β ( z ) γ ( z ) ] 1 2 , τ c ( z ) = τ 0 ( c 0 τ 0 + b 0 ) D ( z ) ,
c ( z ) = c 0 1 c 0 D ( z ) , b ( z ) = b 0 1 c 0 D ( z ) , d ( z ) = ( b 0 2 2 ) D ( z ) 1 c 0 D ( z ) .
g ( z ) = β ( z ) c ( z ) + d dz ln [ β ( z ) γ ( z ) ] .
U B ( χ , ζ ) = aA sech [ a ( χ νζ ) ] e i ( Φ + Θ B )
U D ( χ , ζ ) = u 0 [ cos ( ϕ ) tanh ( Θ D ) + i sin ( ϕ ) ] e i ( u 0 2 ζ + Φ ) .
Θ B ( χ , ζ ) = νχ 2 + a 2 ζ ν 2 4 ,
Θ D ( χ , ζ ) = u 0 cos ( ϕ ) [ χ u 0 ζ sin ( ϕ ) ] .
g ( z ) = c 0 β 1 c 0 βz .
I ( ζ , τ ) = 2 A 2 ( z ) 2 χ 2 ln ( det M ) ,
det M = 1 + e 2 ( η 1 + η 1 * + δ 11 ) + e 2 ( η 2 + η 2 * + δ 22 ) e 2 ( η 1 + η 2 * + δ 12 ) e 2 ( η 1 * + η 2 + δ 12 * ) + K e 4 Re ( η 1 + η 2 ) ,
K = e 2 ( δ 11 + δ 22 ) + e 2( δ 12 + δ 12 * ) 2 e ( δ 11 + δ 22 + δ 12 + δ 12 * ) .
η j = i λ j ( χ + λ j ζ ) , e δ jk = { r j ( 2 v j ) , j = k ; ( r j r k * ) 1 2 ( λ j λ k * ) , j k .
τ τ 0 + ( c 0 τ 0 + b 0 2 μ j w 0 ) D ( z ) 1 c 0 D ( z ) = const .
I j ( χ , ζ ) ~ 4 v j 2 A 2 ( z ) sech 2 [ 2 v j ( χ + 2 μ j ζ ) + Δ j ] ,
Δ j = 1 2 ln [ e 2 δ jj + e 2 ( δ 12 + δ 12 * δ 3 j , 3 j ) 2 e ( δ 12 + δ 12 * + δ jj δ 3 j , 3 j ) ] .
lim z D ( z ) = ,
I j ( τ , z ) ~ 4 v j 2 A 2 ( z ) sech 2 { 2 v j w 0 [ τ + κ j D ( z ) ] + Δ ˜ j } ,

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