Abstract

We consider the nonlinear Schrödinger equation with variable coefficients that describes beam propagation in inhomogeneous graded-index waveguides. By using the direct transformation of variables and functions, we present the exact general bright and dark spatial self-similar solutions. As an application, we discuss the nonlinear tunneling of optical similaritons. The results show that under an integrable condition, the optical waves can similarly pass through the nonlinear barrier or well, and the interaction between the neighboring waves is elastic collision. Under a nonintegrable condition, when they pass through the nonlinear barrier, the optical beams can effectively be compressed for the relatively small value of height of the nonlinear barrier. However, the beam splits into some filaments when the height of the nonlinear barrier is large enough.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  4. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
    [CrossRef]
  5. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  6. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502-4505 (2000).
    [CrossRef] [PubMed]
  7. V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. 8, 418-431 (2002).
    [CrossRef]
  8. 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]
  9. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external pontentials,” Phys. Rev. Lett. 98, 074102 (2007).
    [CrossRef] [PubMed]
  10. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
    [CrossRef]
  13. J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
    [CrossRef]
  14. H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. S. A. Ponomarenko and G. P. Agrawal, “Do solitonlike self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
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    [CrossRef]
  24. G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
    [CrossRef]
  25. A. C. Newell, “Nonlinear tunneling,” J. Math. Phys. 19, 1126-1133 (1978).
    [CrossRef]
  26. D. Anderson, M. Lisak, B. Malomed, and M. Quiroga-Teixeiro, “Tunneling of an optical soliton through a fiber junction,” J. Opt. Soc. Am. B 11, 2380-2384 (1994).
    [CrossRef]
  27. V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
    [CrossRef]

2008 (1)

2007 (6)

L. Wu, J. F. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. 9, 69-69 (2007).
[CrossRef]

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express 15, 2963-2973 (2007).
[CrossRef] [PubMed]

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

S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659-1661 (2007).
[CrossRef] [PubMed]

H. M. Li and F. Q. Song, “Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity,” Opt. Commun. 277, 174-180 (2007).
[CrossRef]

2006 (3)

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

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]

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

2005 (2)

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]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

2004 (1)

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

2003 (1)

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

2002 (2)

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

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

2001 (3)

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, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

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

2000 (2)

V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502-4505 (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]

1994 (1)

1993 (1)

1992 (1)

1978 (1)

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

Agrawal, G. P.

Anderson, D.

Barenblatt, G. I.

G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U. Press, 1996).

Belyaeva, T. L.

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

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

V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Buckley, J. R.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Chapela, V. M.

V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Clark, W. G.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Desaix, M.

Dudley, J. M.

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
[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]

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]

Hao, R. Y.

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

Harvey, J. D.

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]

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

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
[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 pontentials,” Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

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

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

Ilday, F. Ö.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Jia, S. T.

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

Karlsson, M.

Kruglov, V. I.

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]

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

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
[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]

Li, H. M.

H. M. Li and F. Q. Song, “Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity,” Opt. Commun. 277, 174-180 (2007).
[CrossRef]

Li, L.

L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, “Similaritons in nonlinear optical systems,” Opt. Express 16, 6352-6360 (2008).
[CrossRef] [PubMed]

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

L. Wu, J. F. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. 9, 69-69 (2007).
[CrossRef]

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

Li, Z. H.

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

Lisak, M.

Malomed, B.

Malomed, B. A.

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

Manassah, J.

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]

Mihalache, D.

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[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]

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

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

Persino, J.

V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

Ponomarenko, S. A.

Porsezian, K.

Quiroga-Teixeiro, M.

Quiroga-Teixeiro, M. L.

Serkin, V. N.

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

V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. 8, 418-431 (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 T. L. Belyaeva, “High-energy optical Schrödinger solitons,” JETP Lett. 74, 573-577 (2001).
[CrossRef]

V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

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

Song, F. Q.

H. M. Li and F. Q. Song, “Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity,” Opt. Commun. 277, 174-180 (2007).
[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]

Tian, Q.

Wang, J. F.

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

Wang, L. Y.

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

Wise, F. W.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Wu, L.

L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, “Similaritons in nonlinear optical systems,” Opt. Express 16, 6352-6360 (2008).
[CrossRef] [PubMed]

L. Wu, J. F. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. 9, 69-69 (2007).
[CrossRef]

Yang, G. Y.

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

Zhang, H. F.

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

Zhang, J. F.

L. Wu, J. F. Zhang, L. Li, Q. Tian, and K. Porsezian, “Similaritons in nonlinear optical systems,” Opt. Express 16, 6352-6360 (2008).
[CrossRef] [PubMed]

L. Wu, J. F. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. 9, 69-69 (2007).
[CrossRef]

Zhou, G. S.

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[CrossRef]

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

Chin. Phys. (1)

H. F. Zhang, J. F. Wang, L. Li, S. T. Jia, and G. S. Zhou, “Generation and propagation of subpicosecond pulse train,” Chin. Phys. 16, 449-455 (2007).
[CrossRef]

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

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

J. Math. Phys. (1)

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

J. Opt. Soc. Am. B (3)

JETP Lett. (1)

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

New J. Phys. (1)

L. Wu, J. F. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. 9, 69-69 (2007).
[CrossRef]

Opt. Commun. (5)

J. F. Wang, L. Li, Z. H. Li, G. S. Zhou, D. Mihalache, and B. A. Malomed, “Generation, compression and propagation of pulse trains under higher-order effects,” Opt. Commun. 263, 328-336 (2006).
[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]

G. Y. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, “Cascade compression induced by nonlinear barriers in propagation of optical solitons,” Opt. Commun. 260, 282-287 (2006).
[CrossRef]

V. N. Serkin, V. M. Chapela, J. Persino, and T. L. Belyaeva, “Nonlinear tunneling of temporal and spatial optical solitons through organic thin films and polymeric waveguides,” Opt. Commun. 192, 237-244 (2001).
[CrossRef]

H. M. Li and F. Q. Song, “Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity,” Opt. Commun. 277, 174-180 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. E (2)

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]

L. Y. Wang, L. Li, Z. H. Li, G. S. Zhou, and D. Mihalache, “Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. E 72, 036614 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

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]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external pontentials,” 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 Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502-4505 (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]

Other (2)

G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics (Cambridge U. Press, 1996).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (10)

Fig. 1
Fig. 1

Evolutions of the nonlinear parameter R ( z ) and the gain–loss function G ( z ) in the system given by F ( Z ) = g 1 2 and Eq. (11). (a) β = 5 ; (b) β = 0.5 . Other parameters: g 1 = 0.1 , δ = 8 , and Z 0 = 2 .

Fig. 2
Fig. 2

Evolutions of nonlinear tunneling of the bright and dark self-similar waves in the nonlinear barrier given by Eq. (11) with β = 5 . (a) Bright spatial self-similar waves given by Eq. (9); (b) dark spatial self-similar waves given by Eq. (10). Here, the system parameters are the same as in Fig. 1, and the self-similar parameters are W ( 0 ) = 1 , C 0 ( 0 ) = 0 , C 1 ( 0 ) = 0.4 , and X c ( 0 ) = 0.1 .

Fig. 3
Fig. 3

Evolutions of nonlinear tunneling of the bright and dark self-similar waves in the nonlinear well given by Eq. (11) with β = 0.5 . (a) Bright spatial self-similar waves given by Eq. (9); (b) dark spatial self-similar waves given by Eq. (10). Here, the parameters are the same as in Fig. 2 except for β.

Fig. 4
Fig. 4

(a) Evolution of the peak value of the bright spatial self-similar waves of Eq. (9) for the system of Eq. (14). The parameters r = 1 and g 2 = 0.2 , 0, 0.2 , and 0.3 . (b) Corresponding evolution of the bright self-similar waves with g 2 = 0.2 . The other parameters are the same as in Fig. 1a.

Fig. 5
Fig. 5

Evolution dynamics of the self-similar waves in the system given by Eqs. (11, 15). (a) Bright and (c) dark self-similar waves through the nonlinear barrier ( β = 5 ) , and (b) bright and (d) dark self-similar waves through the nonlinear well ( β = 0.5 ) . The location of the nonlinear barrier and well is Z 0 = 3 . The system parameters are β 1 = 0.1 , σ = 10 , and δ = 8 . The wave parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Contour scenarios of the bright self-similar waves in the system given by F ( Z ) = g 1 2 and Eqs. (11, 16). The height of the nonlinear well or barrier β equals to (a) 0.5 , (b) 0, (c) 1, (d) 3, (e) 5, and (f) 8, respectively. The other parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Contour scenarios of the neighboring bright self-similar waves in the system given by F ( Z ) = g 1 2 and Eq. (14). The initial central position X c ( 0 ) equals to (a) 2, (b) 3, (c) 4, (d) 5, (e) 6, and (f) 7, respectively. The system parameters are g 1 = 0.1 , r = 1 , g 2 = 0.2 , β = 5 , δ = 8 , and Z 0 = 2 . The other self-similar wave parameters are W ( 0 ) = 1 , C 0 ( 0 ) = 0 , and C 1 ( 0 ) = 0 .

Fig. 8
Fig. 8

Interaction scenario of the neighboring bright self-similar waves in the system given by Eqs. (11, 15) with the initial separation parameter X c ( 0 ) = 7 . The system parameters are β 1 = 0.1 , σ = 10 , β = 5 , δ = 8 , and Z 0 = 3 . The other self-similar wave parameters are the same as in Fig. 7.

Fig. 9
Fig. 9

Interaction scenarios of the neighboring bright self-similar waves under the nonintegrable condition of Eq. (16) for the different parameters β. (a) β = 0.5 , (b) β = 0 , (c) β = 1 , (d) β = 3 , (e) β = 5 , and (f) β = 8 . Here X c ( 0 ) = 7 , and the other parameters are the same as in Fig. 7.

Fig. 10
Fig. 10

Interaction scenarios of the neighboring bright self-similar waves under the nonintegrable condition of Eq. (16) for the different initial separation parameters X c ( 0 ) . (a) X c ( 0 ) = 2 , (b) X c ( 0 ) = 3 , (c) X c ( 0 ) = 4 , (d) X c ( 0 ) = 5 , (e) X c ( 0 ) = 6 , and (f) X c ( 0 ) = 7 . Here β = 3 , and the other parameters are the same as in Fig. 7.

Equations (26)

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n ( z , x ) = n 0 + n 1 F ( z ) x 2 + n 2 R ( z ) I ( z , x ) ,
i u z + 1 2 k 0 2 u x 2 + k 0 n 1 n 0 F ( z ) x 2 u + k 0 n 2 n 0 R ( z ) u 2 u = i g ( z ) 2 u ,
i U Z + 1 2 2 U X 2 + F ( Z ) 2 X 2 U + σ R ( Z ) U 2 U = i G ( Z ) 2 U ,
U ( X , Z ) = 1 W ( Z ) R ( Z ) Φ ( ξ , η ) exp [ i B ( X , Z ) ] ,
ξ = X X c ( Z ) W ( Z ) , η = η ( Z ) .
B ( X , Z ) = C 2 ( Z ) X 2 + C 1 ( Z ) X + C 0 ( Z )
d 2 W d Z 2 = F W ,
η ( Z ) = 0 Z d ς W 2 ( ς ) ,
X c ( Z ) = W ( 0 ) C 1 ( 0 ) W 0 Z d ς W 2 ( ς ) + X c ( 0 ) W ( 0 ) W ,
C 0 ( Z ) = C 0 ( 0 ) C 1 2 ( 0 ) W 2 ( 0 ) 2 0 Z d ς W 2 ( ς ) ,
C 1 ( Z ) = C 1 ( 0 ) W ( 0 ) W ,
C 2 ( Z ) = 1 2 W d W d Z ,
i Φ η + 1 2 2 Φ ξ 2 + σ Φ 2 Φ = 0 ,
G = 1 W d W d Z 1 R d R d Z .
U B ( X , Z ) = 1 W ( Z ) R ( Z ) sech [ X X c ( Z ) W ( Z ) ] × exp [ i B ( X , Z ) + i 1 2 η ( Z ) ] .
U D ( X , Z ) = 1 W ( Z ) R ( Z ) tanh [ X X c ( Z ) W ( Z ) ] × exp [ i B ( X , Z ) i η ( Z ) ] ,
R ( Z ) = 1 + β sech 2 [ δ ( Z Z 0 ) ] ,
W ( Z ) = W ( 0 ) e g 1 Z ,
G ( Z ) = g 1 + 2 β δ sech 2 [ δ ( Z Z 0 ) ] tanh [ δ ( Z Z 0 ) ] 1 + β sech 2 [ δ ( Z Z 0 ) ] .
R ( Z ) = r e g 2 Z + β sech 2 [ δ ( Z Z 0 ) ] ,
G ( Z ) = g 1 + g 2 r e g 2 Z r e g 2 Z + β sech 2 [ δ ( Z Z 0 ) ] + 2 δ β sech 2 [ δ ( Z Z 0 ) ] tanh [ δ ( Z Z 0 ) ] r e g 2 Z + β sech 2 [ δ ( Z Z 0 ) ] .
F ( Z ) = σ 2 β 1 sin ( σ Z ) [ 1 + β 1 sin ( σ Z ) ] ,
G ( Z ) = 2 β δ sech 2 [ δ ( Z Z 0 ) ] tanh [ δ ( Z Z 0 ) ] 1 + β sech 2 [ δ ( Z Z 0 ) ] σ β 1 cos ( σ Z ) 1 + β 1 sin ( σ Z ) ,
W ( Z ) = W ( 0 ) [ 1 + β 1 sin ( σ Z ) ] .
G = g 1 .
U int ( X , 0 ) = 1 W ( 0 ) R ( 0 ) { sech [ X X c ( 0 ) W ( 0 ) ] + sech [ X + X c ( 0 ) W ( 0 ) ] } exp [ i B ( X , 0 ) ] .

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