Abstract

By using the lens-type transformation, exact soliton and quasi-soliton similaritons are found in (1+1), (2+1) and (3+1)-dimensional nonlinear Schrödinger equations in the context of nonlinear optical fiber amplifiers and graded-index waveguide amplifiers. The novel analytical and numerical results show that, in addition to the exact solitonic optical waves, quasi-solitonic optical waves with Gaussian, parabolic, vortex and ring soliton profiles can evolve exact self-similarly without any radiation.

© 2008 Optical Society of America

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  1. 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]
  2. 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, 1753-1755 (2000).
    [CrossRef]
  3. 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, 461-468 (2002).
    [CrossRef]
  4. G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (2005).
    [CrossRef]
  5. 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 (2005).
    [CrossRef]
  6. V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502 (2000).
    [CrossRef] [PubMed]
  7. V. N. Serkin and A. Hasegawa, "Soliton management in the nonlinear Schr¨odinger equation model with varying dispersion, nonlinearity, and gain," JETP Letters,  72, 89-92 (2000).
    [CrossRef]
  8. V. N. Serkin and A. Hasegawa, "Exactly integrable nonlinear Schr¨odinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements," IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
    [CrossRef]
  9. 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 (2003).
    [CrossRef] [PubMed]
  10. V. M. Perez-Garcıa, P. J. Torres, and V. V. Konotop, "Similarity transformations for nonlinear Schr¨odinger equations with time-dependent coefficients," Physica D 221, 31-36 (2006).
    [CrossRef]
  11. 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]
  12. 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]
  13. S. A. Ponomarenko and G. P. Agrawal, "Optical similaritons in nonlinear waveguides," Opt. Lett. 32, 1659-1661 (2007).
    [CrossRef] [PubMed]
  14. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous solitons in external potentials," Phys. Rev. Lett. 98, 074102 (2007).
    [CrossRef] [PubMed]
  15. T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).
  16. C. Hernandez Tenorio, E. Villagran. Vargas, V. N. Serkin, M. Ag¨uero Granados, T. L. Belyaeva, R. Pe?na Moreno, and L. Morales Lara, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: II. Dark solitons," Quantum Electron. 10, 929-937 (2005).
    [CrossRef]
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    [CrossRef]
  22. 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]
  23. S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
    [CrossRef]
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    [CrossRef]
  27. G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
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  28. J.-K. Xue, "Controllable compression of bright soliton matter waves", J. Phys. B: At. Mol. Opt. Phys. 38, 3841 (2005).
    [CrossRef]
  29. L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
    [CrossRef]
  30. 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 (2007).
    [CrossRef]
  31. L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
    [CrossRef]
  32. 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 (1972).

2008

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[CrossRef]

2007

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 (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]

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

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

2006

V. M. Perez-Garcıa, P. J. Torres, and V. V. Konotop, "Similarity transformations for nonlinear Schr¨odinger equations with time-dependent coefficients," Physica D 221, 31-36 (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]

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]

L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
[CrossRef]

2005

J.-K. Xue, "Controllable compression of bright soliton matter waves", J. Phys. B: At. Mol. Opt. Phys. 38, 3841 (2005).
[CrossRef]

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

C. Hernandez Tenorio, E. Villagran. Vargas, V. N. Serkin, M. Ag¨uero Granados, T. L. Belyaeva, R. Pe?na Moreno, and L. Morales Lara, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: II. Dark solitons," Quantum Electron. 10, 929-937 (2005).
[CrossRef]

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (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 (2005).
[CrossRef]

2003

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 (2003).
[CrossRef] [PubMed]

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[CrossRef]

2002

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, 461-468 (2002).
[CrossRef]

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

2000

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, 1753-1755 (2000).
[CrossRef]

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

V. N. Serkin and A. Hasegawa, "Soliton management in the nonlinear Schr¨odinger equation model with varying dispersion, nonlinearity, and gain," JETP Letters,  72, 89-92 (2000).
[CrossRef]

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
[CrossRef]

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

1997

1972

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 (1972).

Agrawal, G. P.

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]

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

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
[CrossRef]

Aguero, G. M.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

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]

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

Bullough, R. K.

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

Chang, G.

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Chen, G.

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[CrossRef]

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 (2005).
[CrossRef]

Dudley, J. M.

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]

Frantzeskakis, D. J.

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[CrossRef]

Galvanauskas, A.

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (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 (2005).
[CrossRef]

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 (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, 461-468 (2002).
[CrossRef]

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, 1753-1755 (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 (2007).
[CrossRef] [PubMed]

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

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

V. N. Serkin and A. Hasegawa, "Soliton management in the nonlinear Schr¨odinger equation model with varying dispersion, nonlinearity, and gain," JETP Letters,  72, 89-92 (2000).
[CrossRef]

S. Kumar and A. Hasegawa, "Quasi-soliton propagation in dispersion-managed optical fibers," Opt. Lett. 6, 372-374 (1997).
[CrossRef]

Hernandez, T. C.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

Hernandez Tenorio, C.

C. Hernandez Tenorio, E. Villagran. Vargas, V. N. Serkin, M. Ag¨uero Granados, T. L. Belyaeva, R. Pe?na Moreno, and L. Morales Lara, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: II. Dark solitons," Quantum Electron. 10, 929-937 (2005).
[CrossRef]

Inoue, T.

Kevrekidis, P. G.

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[CrossRef]

Konotop, V. V.

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[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 (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, 461-468 (2002).
[CrossRef]

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, 1753-1755 (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]

Kumar, S.

Li, L.

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[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 (2007).
[CrossRef]

Lindberg, M.

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[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 (2005).
[CrossRef]

Norris, T. B.

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Ozeki, Y.

Peacock, A. C.

Ponomarenko, S. A.

Raghavan, S.

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
[CrossRef]

Rapti, Z.

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[CrossRef]

Rybin, A. V.

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

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]

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

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

V. N. Serkin and A. Hasegawa, "Soliton management in the nonlinear Schr¨odinger equation model with varying dispersion, nonlinearity, and gain," JETP Letters,  72, 89-92 (2000).
[CrossRef]

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502 (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 (1972).

Theocharis, G.

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[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.

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[CrossRef]

Timonen, J.

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

Varzugin, G. G.

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

Villagran, E.

C. Hernandez Tenorio, E. Villagran. Vargas, V. N. Serkin, M. Ag¨uero Granados, T. L. Belyaeva, R. Pe?na Moreno, and L. Morales Lara, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: II. Dark solitons," Quantum Electron. 10, 929-937 (2005).
[CrossRef]

Villargan, V. E.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

Winful, H. G.

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Wu, L.

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[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 (2007).
[CrossRef]

L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
[CrossRef]

Xue, J.-K.

J.-K. Xue, "Controllable compression of bright soliton matter waves", J. Phys. B: At. Mol. Opt. Phys. 38, 3841 (2005).
[CrossRef]

Yang, Q.

L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
[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 (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 (1972).

Zhang, J.-f.

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[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 (2007).
[CrossRef]

L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

J. Opt. Soc. Am. B

J. Phys. A: Math. Gen.

L. Wu, Q. Yang, and J.-f. Zhang, "Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schr¨odinger equation in plasma," J. Phys. A: Math. Gen. 39, 11947-11953 (2006).
[CrossRef]

JETP Letters

V. N. Serkin and A. Hasegawa, "Soliton management in the nonlinear Schr¨odinger equation model with varying dispersion, nonlinearity, and gain," JETP Letters,  72, 89-92 (2000).
[CrossRef]

New J. Phys.

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 (2007).
[CrossRef]

L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, "Controllable exact self-similar evolution of the Bose-Einstein condensate", New J. Phys. 10, 023021 (2008).
[CrossRef]

Opt. Commun.

S. Raghavan and G. P. Agrawal, "Spatiotemporal solitons in inhomogeneous nonlinear media," Opt. Commun. 180, 377-382 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Phys.

J.-K. Xue, "Controllable compression of bright soliton matter waves", J. Phys. B: At. Mol. Opt. Phys. 38, 3841 (2005).
[CrossRef]

Phys. Rev. A

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, "Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions," Phys. Rev. A 67, 063610 (2003).
[CrossRef]

Phys. Rev. E

A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, "Similarity solutions and collapse in the attractive Gross-Pitaevskii equation," Phys. Rev. E 62, 6224 (2000).
[CrossRef]

G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, "Self-similar parabolic beam generation and propagation," Phys. Rev. E 72, 016609 (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 (2005).
[CrossRef]

Phys. Rev. Lett.

V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schr¨odinger equation model," Phys. Rev. Lett. 85, 4502 (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. 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 (2003).
[CrossRef] [PubMed]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, "Nonautonomous solitons in external potentials," Phys. Rev. Lett. 98, 074102 (2007).
[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]

Physica D

V. M. Perez-Garcıa, P. J. Torres, and V. V. Konotop, "Similarity transformations for nonlinear Schr¨odinger equations with time-dependent coefficients," Physica D 221, 31-36 (2006).
[CrossRef]

Quantum Electron.

T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Aguero, T. L. Belyaeva, M. R. Pena, and L. L. Morales, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: I. Bright solitons," Quantum Electron. 9, 778-786 (2005).

C. Hernandez Tenorio, E. Villagran. Vargas, V. N. Serkin, M. Ag¨uero Granados, T. L. Belyaeva, R. Pe?na Moreno, and L. Morales Lara, "Dynamics of solitons in the model of nonlinear Schr¨odinger equation with an external harmonic potential: II. Dark solitons," Quantum Electron. 10, 929-937 (2005).
[CrossRef]

Sov. Phys. JETP

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 (1972).

Other

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (University of Cambridge, Cambridge, 2001).
[CrossRef]

C. Sulem and P. L. Sulem, The Nonlinear Schrodinger Equation (Springer-Verlag, New York, 1999).

Y. R. Shen, The principles of nonlinear optics (John Wiley and Sons, 2003).

P. G. Drazin and R. S. Jonson, Solitons: an Introduction (Cambridge University Press, 1988).

F. Calogero and A. Degasperis, Spectral Transform and Solitons (North -Holland, Amsterdam, 1982).

G. L. Lamb, Elements of Soliton Theory (Cambridge University Press, 1980).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Self-similar evolution of initial Gaussian beams, (b) inhomogeneous parameters f(z), (c) the amplitudes and (d) the widths (defined as 2 x 2 u 2 d x u 2 d x ) of Gaussian similaritons as functions of propagation distance z. The solid lines are theoretical predictions, dots and open circles are numerical simulations with K=13.2 for σ=1 and K=18.8 for σ=-1, respectively.

Fig. 2.
Fig. 2.

The theoretical prediction of the propagation of parabolic similaritons (solid lines) inside the graded-index waveguide amplifier with gain parameter g=g 1/3 and inhomogeneous parameter f is confirmed by numerical simulations (open circles), where the parabolic beam (at z=24) is generated by injecting a Gaussian beam (z=0) into the homogeneous planar waveguide with gain parameter g 1=0.3 (dashed-lines show its evolution into parabolic beam). From bottom to top, the propagation distance z is 0, 4.8, 9.6, 14.4, 19.2, 24, 26, 28, 30, 32 and 34, respectively.

Fig. 3.
Fig. 3.

Exact self-similar evolution of vortex beam in two-dimensional graded-index waveguide with σ=-1 and f(z)=1-2sech2(z)-K/sech4(z) such that =sech(z). From bottom to top, the six radial profiles correspond to the propagation distance z=0,0.4,0.8,1.2,1.6,2.0, respectively. Here K=0.01.

Fig. 4.
Fig. 4.

Exact self-similar evolution of ring soliton in two-dimensional grade-index waveguide with σ=1 and f(z)=1-Ksech4(z) such that =1/sech(z). Here K=0.01.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

i u z + 1 2 u xx + σ u 2 u + f ( z ) x 2 2 u = i g ( z ) 2 u ,
i ψ Z + β ( Z ) 2 ψ TT + γ ( Z ) ψ 2 ψ = i G ( Z ) 2 ψ ,
i u z + 1 2 ( u xx + u yy ) + σ u 2 u + f ( z ) x 2 + y 2 2 u = i g ( z ) 2 u ,
i u z + 1 2 ( u xx + u yy ) + δ 2 u ττ + σ u 2 u + f ( z ) x 2 + y 2 2 u = i g ( z ) 2 u ,
u ( z , x ) = exp [ 0 z g ( z ) 2 d z ] ( z ) U ( Z , X ) exp [ i z 2 x 2 ] ,
i U Z + 1 2 U XX + σ U 2 U = K X 2 2 U ,
u ( z , x ) = 1 ( z ) S [ x ( z ) ] exp [ i μ 0 z d z ( z ) 2 i g ( z ) 2 x 2 ] .
f = g 2 g z K exp [ 4 0 z g ( z ) d z ] .
g 2 g z = K exp [ 4 0 z g ( z ) d z ] .
g 2 g Z β = K exp [ 4 0 Z g ( z ) β ( z ) d z ] .
β β ZZ β Z 2 = K ,
G + γ Z γ = Z + c 2 ( Z + c 2 ) 2 + c 1 ,
i U Z + 1 2 ( U XX + U YY ) + σ U 2 U = K X 2 + Y 2 2 U ,
u ( z , x , y ) = 1 U ( Z , X , Y ) exp [ i z ( x 2 + y 2 ) 2 ] ,
u ( z , x , y , τ ) = 1 U ( Z , X , Y , T ) exp [ i g 2 ( x 2 + y 2 + τ 2 δ ) ] ,
i U Z + 1 2 ( U XX + U YY + δ U TT ) + σ U 2 U = K XY 2 ( X 2 + Y 2 ) + K T 2 T 2 ,
K XY = ( g z + g 2 f ) 4 , K T = ( g z + g 2 ) 4 .
g = z + c 2 ( z + c 2 ) 2 + c 1 , f = K T K XY 2 ln [ ( z + c 2 ) 2 + c 1 ] ,

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