Abstract

The propagation of optical pulses inside dispersion-managed fibers is considered. It is found that the chirped compact parabolic pulse can propagate inside the dispersion-managed fibers self-similarly. Such a finite-width pulse can be served as the background for the propagation and interaction of dark similaritons. Approximate but highly accurate analytical methods are proposed to describe the interaction dynamics of multiple dark similaritons on the self-similar compact parabolic background.

©2009 Optical Society of America

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References

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  1. N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
    [Crossref]
  2. Shiva Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. 22, 372–374 (1997).
    [Crossref] [PubMed]
  3. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrodinger equation Model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
    [Crossref] [PubMed]
  4. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
    [Crossref]
  5. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902–113905 (2003).
    [Crossref] [PubMed]
  6. S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
    [Crossref]
  7. Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. 31, 1606–1608 (2006).
    [Crossref] [PubMed]
  8. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102–074105 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  10. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  13. L. Wu, L. Li, and J.-F. Zhang, “Controllable generation and propagation of asymptotic parabolic optical waves in graded-index waveguide amplifiers,” Phys. Rev. A 78, 013838 (2008).
    [Crossref]
  14. C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15, 15824–15835 (2007).
    [Crossref] [PubMed]
  15. T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion,” Opt. Lett. 29, 498–500 (2004).
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  17. S. A. Ponomarenko and G. P. Agrawal, “Nonlinear interaction of two or more similaritons in loss- and dispersion-managed fibers,” J. Opt. Soc. Am. B 25, 983–989 (2008).
    [Crossref]
  18. V. A. Bogatyrevet al, “A single-mode fiber with chromatic dispersion varying along thelength,” IEEE J. Lightwave Technol. 9, 561 (1991).
    [Crossref]
  19. S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Comblike dispersion-profiled fiber for soliton pulse train generation,” Opt. Lett. 19, 539–541 (1994).
    [Crossref] [PubMed]
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    [Crossref]
  21. K. R. Tamura and M. Nakazawa, “54-fs, 10-GHz soliton generation from a polarization-maintaining dispersion-flattened dispersion-decreasing fiber pulse compressor,” Opt. Lett. 26, 762–764 (2001).
    [Crossref]
  22. V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE Selected Topics in Quantum Electronics 8, 418–432 (2002)
    [Crossref]
  23. V. N. Serkin, M. Matsumoto, and T. L. Belyaeva, “Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159–171 (2001).
    [Crossref]
  24. L. Li, X. S. Zhao, and Z. Y. Xu, “Dark solitons on an intense parabolic background in nonlinear waveguides,” Phys. Rev. A 78, 063833 (2008).
    [Crossref]
  25. T. Busch and J. R. Anglin, “Motion of dark solitons in trapped bose-einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
    [Crossref] [PubMed]
  26. C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
    [Crossref]
  27. L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
    [Crossref]
  28. Y. S. Kivshar and W. Krolikowski, “Lagrangian approach for dark solitons,” Opt. Commun. 114, 353–362 (1995).
    [Crossref]
  29. It happens when the depth of the dark similariton is shallow, i.e., the velocity of dark similartion in large).

2008 (4)

L. Wu, L. Li, and J.-F. Zhang, “Controllable generation and propagation of asymptotic parabolic optical waves in graded-index waveguide amplifiers,” Phys. Rev. A 78, 013838 (2008).
[Crossref]

S. A. Ponomarenko and G. P. Agrawal, “Nonlinear interaction of two or more similaritons in loss- and dispersion-managed fibers,” J. Opt. Soc. Am. B 25, 983–989 (2008).
[Crossref]

L. Li, X. S. Zhao, and Z. Y. Xu, “Dark solitons on an intense parabolic background in nonlinear waveguides,” Phys. Rev. A 78, 063833 (2008).
[Crossref]

L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
[Crossref]

2007 (4)

2006 (3)

S. A. Ponomarenko and G. P. Agrawal, “Do Solitonlike Self-Similar Waves Exist in Nonlinear Optical Media?” Phys. Rev. Lett. 97, 013901–013904 (2006).
[Crossref] [PubMed]

Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. 31, 1606–1608 (2006).
[Crossref] [PubMed]

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

2005 (2)

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
[Crossref]

2004 (1)

2003 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902–113905 (2003).
[Crossref] [PubMed]

2002 (1)

V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE Selected Topics in Quantum Electronics 8, 418–432 (2002)
[Crossref]

2001 (2)

V. N. Serkin, M. Matsumoto, and T. L. Belyaeva, “Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159–171 (2001).
[Crossref]

K. R. Tamura and M. Nakazawa, “54-fs, 10-GHz soliton generation from a polarization-maintaining dispersion-flattened dispersion-decreasing fiber pulse compressor,” Opt. Lett. 26, 762–764 (2001).
[Crossref]

2000 (4)

T. Busch and J. R. Anglin, “Motion of dark solitons in trapped bose-einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrodinger equation Model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[Crossref] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
[Crossref]

1997 (1)

1995 (1)

Y. S. Kivshar and W. Krolikowski, “Lagrangian approach for dark solitons,” Opt. Commun. 114, 353–362 (1995).
[Crossref]

1994 (1)

1991 (1)

V. A. Bogatyrevet al, “A single-mode fiber with chromatic dispersion varying along thelength,” IEEE J. Lightwave Technol. 9, 561 (1991).
[Crossref]

1965 (1)

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[Crossref]

Agrawal, G. P.

Anglin, J. R.

T. Busch and J. R. Anglin, “Motion of dark solitons in trapped bose-einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

Barviau, B.

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102–074105 (2007).
[Crossref] [PubMed]

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

V. N. Serkin, M. Matsumoto, and T. L. Belyaeva, “Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159–171 (2001).
[Crossref]

Billet, C.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

Bogatyrev, V. A.

V. A. Bogatyrevet al, “A single-mode fiber with chromatic dispersion varying along thelength,” IEEE J. Lightwave Technol. 9, 561 (1991).
[Crossref]

Busch, T.

T. Busch and J. R. Anglin, “Motion of dark solitons in trapped bose-einstein condensates,” Phys. Rev. Lett. 84, 2298–2301 (2000).
[Crossref] [PubMed]

Chen, S.

S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
[Crossref]

Chernikov, S. V.

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597 (2007).
[Crossref]

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
[Crossref]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

Ferriere, R.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

Finot, C.

L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
[Crossref]

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597 (2007).
[Crossref]

C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15, 15824–15835 (2007).
[Crossref] [PubMed]

Granados, M. A.

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

Guo, D.-S.

S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
[Crossref]

Guryanov, A.

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902–113905 (2003).
[Crossref] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102–074105 (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 managements,” IEEE Selected Topics in Quantum Electronics 8, 418–432 (2002)
[Crossref]

V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrodinger equation Model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[Crossref] [PubMed]

Shiva Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. 22, 372–374 (1997).
[Crossref] [PubMed]

Hirooka, T.

Inoue, T.

Kashyap, R.

Kibler, B.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

Kivshar, Y. S.

Y. S. Kivshar and W. Krolikowski, “Lagrangian approach for dark solitons,” Opt. Commun. 114, 353–362 (1995).
[Crossref]

Krolikowski, W.

Y. S. Kivshar and W. Krolikowski, “Lagrangian approach for dark solitons,” Opt. Commun. 114, 353–362 (1995).
[Crossref]

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902–113905 (2003).
[Crossref] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
[Crossref]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

Kruskal, M. D.

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[Crossref]

Kumar, Shiva

Lacourt, P. A.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

Lara, L. M.

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

Larger, L.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comblike profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006).
[Crossref]

Li, L.

L. Li, X. S. Zhao, and Z. Y. Xu, “Dark solitons on an intense parabolic background in nonlinear waveguides,” Phys. Rev. A 78, 063833 (2008).
[Crossref]

L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
[Crossref]

L. Wu, L. Li, and J.-F. Zhang, “Controllable generation and propagation of asymptotic parabolic optical waves in graded-index waveguide amplifiers,” Phys. Rev. A 78, 013838 (2008).
[Crossref]

Lu, P.

S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
[Crossref]

Matsumoto, M.

V. N. Serkin, M. Matsumoto, and T. L. Belyaeva, “Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159–171 (2001).
[Crossref]

Millot, G.

Moreno, R. P.

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

Nakazawa, M.

Ozeki, Y.

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902–113905 (2003).
[Crossref] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 175–177 (2000).
[Crossref]

Ponomarenko, S. A.

Porsezian, K.

L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
[Crossref]

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597 (2007).
[Crossref]

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102–074105 (2007).
[Crossref] [PubMed]

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrodinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE Selected Topics in Quantum Electronics 8, 418–432 (2002)
[Crossref]

V. N. Serkin, M. Matsumoto, and T. L. Belyaeva, “Bright and dark solitary nonlinear Bloch waves in dispersion managed fiber systems and soliton lasers,” Opt. Commun. 196, 159–171 (2001).
[Crossref]

V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrodinger equation Model,” Phys. Rev. Lett. 85, 4502–4505 (2000).
[Crossref] [PubMed]

Sysoliatin, A.

Tamura, K. R.

Taylor, J. R.

Tenorio, C. H.

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar Propagation and Amplification of Parabolic Pulses in Optical Fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[Crossref] [PubMed]

Vargas, E. V.

C. H. Tenorio, E. V. Vargas, V. N. Serkin, M. A. Granados, T. L. Belyaeva, R. P. Moreno, and L. M. Lara, “Dynamics of solitons in the model of nonlinear Schrodinger equation with an external harmonic potential: Part II. Dark solitons,” Quantum Electronics,  35, 929–937 (2005).
[Crossref]

Wabnitz, S.

Wu, L.

L. Wu, J. F. Zhang, L. Li, C. Finot, and K. Porsezian, “Similaritons interaction in nonlinear graded-index waveguide amplifiers,” Phys. Rev. A 78, 053807 (2008).
[Crossref]

L. Wu, L. Li, and J.-F. Zhang, “Controllable generation and propagation of asymptotic parabolic optical waves in graded-index waveguide amplifiers,” Phys. Rev. A 78, 013838 (2008).
[Crossref]

Xu, Z. Y.

L. Li, X. S. Zhao, and Z. Y. Xu, “Dark solitons on an intense parabolic background in nonlinear waveguides,” Phys. Rev. A 78, 063833 (2008).
[Crossref]

Yi, L.

S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E 72, 016622–016606 (2005).
[Crossref]

Zabusky, N. J.

N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15, 240–243 (1965).
[Crossref]

Zhang, J. F.

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It happens when the depth of the dark similariton is shallow, i.e., the velocity of dark similartion in large).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Boston, 2001).

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Figures (4)

Fig. 1.
Fig. 1. Interaction of two dark similaritons on the self-similar compact parabolic background. The initial parameters for the first dark soliton (left) are Q 1(0) = −5,ϕ 1(0) = 0, that of the second (right) are Q 2(0) =5,ϕ 2(0) =π/8. Other parameters are A=2, K = 0.01, C = −0.1, tc (0) = 1 and b = 0.1. Red and blue lines represents the theoretical predictions and numerical simulations, respectively.
Fig. 2.
Fig. 2. Evolution of three dark similaritons on the self-similar compact parabolic background (contour plot of ∣u2). The red, blue and green lines represent the trajectories of dark similaritons for Q 1 (0) = 5,ϕ 1(0) = π/8, Q 2(0) = 0,ϕ 2(0) = − π/8, and Q 3(0) = −5,ϕ 3(0) = 0, respectively. Other parameters are the same with that in Fig. 1. Inset: the comparison between the numerical/theoretical (blue/red lines) results.
Fig. 3.
Fig. 3. Evolution of the dark similaritons, which initially locate at t = −1, with ϕ(0) = 0 in (a) and ϕ(0) = −π/5 in (b). From tom to bottom, z =30, 27, 24, 21 and 18. The black and color lines are the results of theoretical predictions and numerical simulations, respectively. The parameters used are A = 2,Tmax = 30, Wz (0) = 0, tc (0) = b = 0, and β(z) = exp(−0.01z).
Fig. 4.
Fig. 4. Evolution of two dark similaritons on the self-similar compact parabolic background (contour plot of ∣u2). The red and blue lines represent the trajectories of dark similaritons for t r1 (0) = 5,ϕ 1(0) = π/10 and t r2(0) = −5,ϕ 2(0) = −π/10, respectively. The background parameter are the same with that in Fig. 3, and β = 1 − 0.01z. Inset: the comparison between the numerical/theoretical(blue/red lines) results.

Equations (15)

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iuzβ(z)2utt+γ|u|2u=0.
u(z,t)=1WU(Z,T)exp(iWz2t2),
iUz+v2UTTU2U=K2T2U ,
W2(Wzzβ2βWz)=.
β(z)=cosh(δz)+Csinh(δz)/δ,
Ub(Z,T)=A2KT2/2exp(iA2Z),
U=Ub{cosϕtanh[Acosϕ(Tvq)]+isinϕ},
Qzz+Kv2Q=vZ2vQZ,
u2=A2W(z)2 {1[ttc(z)]2Tmax2W(z)2} {1cosϕ2sech2[Acosϕttc(z)tr(z)W(z)β(z)]} .
trzz=(WzzWWz2W2+WzβWβz2βW2Wz2W3)tr+12(βzβ+WzW)trzF(tr,trz),
trzz=K2β2tr+βzβtrz.
sinϕ=trWz/WtrzAβ/W.
U=Ubi=1N{cosϕitanh[Avcosϕi(TQi)]+isinϕi}
Fij=2A3vsech2(AQij/v)tanh(AQij/v)
trizz=F(tri,triz)+j=1N2A3vWsech2(Δ)tanh(Δ),

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