Abstract

We present exact asymptotic similariton solutions of the generalized nonlinear Schrödinger equation (NLSE) with gain or loss terms for a normal-dispersion fiber amplifier with dispersion, nonlinearity, and gain profiles that depend on the propagation distance. Our treatment is based on the mapping of the NLSE with varying parameters to the NLSE with constant dispersion and nonlinearity coefficients and an arbitrary varying gain function. We formulate an effective procedure that leads directly, under appropriate conditions, to a wide range of exact asymptotic similariton solutions of NLSE demonstrating self-similar propagating regimes with linear chirp.

© 2006 Optical Society of America

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  1. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Taixeiro, 'Wave-breaking-free pulses in nonlinear optical fibers,' J. Opt. Soc. Am. B 10, 1185-1190 (1993).
    [CrossRef]
  2. 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]
  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, 461-469 (2002).
    [CrossRef]
  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, 1753-1755 (2000).
    [CrossRef]
  5. K. Tamura and M. Nakazawa, 'Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,'Opt. Lett. 21, 68-70 (1996).
    [CrossRef] [PubMed]
  6. C. Finot, G. Millot, C. Billet, and J. M. Dudley, 'Experimental generation of parabolic pulses via Raman amplification in optical fiber,' Opt. Express 11, 1547-1552 (2003).
    [CrossRef] [PubMed]
  7. C. Finot, S. Pitois, G. Millot, C. Billet, and J. M. Dudley, 'Numerical and experimental study of parabolic pulses generated via Raman scattering and gain bandwith,' IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
    [CrossRef]
  8. C. Finot, G. Millot, and J. M. Dudley, 'Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers,' Opt. Lett. 29, 2533-2535 (2004).
    [CrossRef] [PubMed]
  9. C. Billet, John Dudley, N. Joly, and J. Knight, 'Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550nm,' Opt. Express 13, 3236-3241 (2005).
    [CrossRef] [PubMed]
  10. C. Finot, F. Parmigiani, P. Petropoulos, and D. Richardson, 'Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,' Opt. Express 14, 3161-3170 (2006).
    [CrossRef] [PubMed]
  11. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, 'Exact solutions of generalized nonlinear Schrödinger equation with distributed coefficients,' Phys. Rev. E 71, 056619 (2005).
    [CrossRef]
  12. D. Méchin, S. H. Im, V. Kruglov, and J. Harvey, 'Experimental demonstration of similariton pulse compression in a comblike dispersion-decreasing fiber amplifier,' Opt. Lett. 31, 2106-2108 (2006).
    [CrossRef] [PubMed]
  13. J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. J. Fuchs, E. B. Kley, H. Zellmer, and A. Tünnermann, 'High power femtosecond Yb-doped fiber amplifier,' Opt. Express 10, 628-638 (2002).
    [PubMed]
  14. A. Malinowski, A. Piper, J. H. Price, K. Furusawa, Y. Jeong, J. Nilsson, and D. J. Richardson, 'Ultrashort-pulse Yb3+-fiber-based laser and amplifier system producing >25W average power,' Opt. Lett. 29, 2073-2075 (2004).
    [CrossRef] [PubMed]
  15. 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]
  16. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, 'Generation of a train of fundamental solitons at a high repetition rate in optical fibers,' Opt. Lett. 14, 1008-1010 (1989).
    [CrossRef] [PubMed]
  17. 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).
    [CrossRef] [PubMed]
  18. V. I. Kruglov, D. Méchin, and J. D. Harvey, 'Self-similar solutions of the generalized Schrödinger equation with distributed coefficients,' Opt. Express 12, 6198-6207 (2004).
    [CrossRef] [PubMed]
  19. V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
    [CrossRef]
  20. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  21. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).
  22. J. D. Moores, 'Nonlinear compression of chirped solitary waves with and without phase modulation,' Opt. Lett. 21, 555-557 (1996).
    [CrossRef] [PubMed]
  23. V. N. Serkin and A. Hasegava, 'Novel soliton solutions of the nonlinear Schrödinger equation model,' Phys. Rev. Lett. 85, 4502-4505 (2000).
    [CrossRef] [PubMed]
  24. V. N. Serkin and A. Hasegava, 'Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: application for soliton dispersion,' IEEE J. Sel. Top. Quantum Electron. 8, 418-431 (2002).
    [CrossRef]
  25. 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]
  26. D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behavior (Chapman & Hall, 1992).

2006 (2)

2005 (3)

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

C. Billet, John Dudley, N. Joly, and J. Knight, 'Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550nm,' Opt. Express 13, 3236-3241 (2005).
[CrossRef] [PubMed]

V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
[CrossRef]

2004 (6)

2003 (2)

C. Finot, G. Millot, C. Billet, and J. M. Dudley, 'Experimental generation of parabolic pulses via Raman amplification in optical fiber,' Opt. Express 11, 1547-1552 (2003).
[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]

2002 (3)

2000 (3)

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]

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

1996 (2)

1993 (1)

1989 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

Anderson, D.

Arrowsmith, D. K.

D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behavior (Chapman & Hall, 1992).

Billet, C.

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]

Chernikov, S. V.

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]

Clausnitzer, T.

Collett, M. J.

V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Desaix, M.

Dianov, E. M.

Dodd, R. K.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).

Dudley, J. M.

Dudley, John

Eilbeck, J. C.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).

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]

Finot, C.

Fuchs, H. J.

Furusawa, K.

Gibbon, J. D.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).

Harvey, J.

Harvey, J. D.

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

V. I. Kruglov, D. Méchin, and J. D. Harvey, 'Self-similar solutions of the generalized Schrödinger equation with distributed coefficients,' Opt. Express 12, 6198-6207 (2004).
[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]

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]

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]

Hasegava, A.

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

Hirooka, T.

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]

Im, S. H.

Jeong, Y.

Joly, N.

Karlsson, M.

Kley, E. B.

Knight, J.

Kruglov, V.

Kruglov, V. I.

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

V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
[CrossRef]

V. I. Kruglov, D. Méchin, and J. D. Harvey, 'Self-similar solutions of the generalized Schrödinger equation with distributed coefficients,' Opt. Express 12, 6198-6207 (2004).
[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]

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]

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]

Limpert, J.

Lisak, M.

Malinowski, A.

Mamyshev, P. V.

Méchin, D.

Millot, G.

Moores, J. D.

Morris, H. C.

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).

Nakazawa, M.

Nilsson, J.

Olsen, M. K.

V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Parmigiani, F.

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, 'Exact solutions of 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]

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]

Petropoulos, P.

Piper, A.

Pitois, S.

C. Finot, S. Pitois, G. Millot, C. Billet, and J. M. Dudley, 'Numerical and experimental study of parabolic pulses generated via Raman scattering and gain bandwith,' IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Place, C. M.

D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behavior (Chapman & Hall, 1992).

Price, J. H.

Prokhorov, A. M.

Quiroga-Taixeiro, M. L.

Richardson, D.

Richardson, D. J.

Schreiber, T.

Serkin, V. N.

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

Tamura, K.

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]

Tünnermann, A.

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]

Zellmer, H.

Zöllner, K.

IEEE J. Sel. Top. Quantum Electron. (2)

C. Finot, S. Pitois, G. Millot, C. Billet, and J. M. Dudley, 'Numerical and experimental study of parabolic pulses generated via Raman scattering and gain bandwith,' IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

V. N. Serkin and A. Hasegava, 'Exactly integrable nonlinear Schrödinger 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 (2)

Opt. Express (5)

Opt. Lett. (8)

A. Malinowski, A. Piper, J. H. Price, K. Furusawa, Y. Jeong, J. Nilsson, and D. J. Richardson, 'Ultrashort-pulse Yb3+-fiber-based laser and amplifier system producing >25W average power,' Opt. Lett. 29, 2073-2075 (2004).
[CrossRef] [PubMed]

D. Méchin, S. H. Im, V. Kruglov, and J. Harvey, 'Experimental demonstration of similariton pulse compression in a comblike dispersion-decreasing fiber amplifier,' Opt. Lett. 31, 2106-2108 (2006).
[CrossRef] [PubMed]

C. Finot, G. Millot, and J. M. Dudley, 'Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers,' Opt. Lett. 29, 2533-2535 (2004).
[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]

K. Tamura and M. Nakazawa, 'Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,'Opt. Lett. 21, 68-70 (1996).
[CrossRef] [PubMed]

E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, 'Generation of a train of fundamental solitons at a high repetition rate in optical fibers,' Opt. Lett. 14, 1008-1010 (1989).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

J. D. Moores, 'Nonlinear compression of chirped solitary waves with and without phase modulation,' Opt. Lett. 21, 555-557 (1996).
[CrossRef] [PubMed]

Phys. Rev. A (1)

V. I. Kruglov, M. K. Olsen, and M. J. Collett, 'Quantum and thermal fluctuations of trapped Bose-Einstein condensates,' Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (4)

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]

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. Hasegava, 'Novel soliton solutions of the nonlinear Schrödinger equation model,' Phys. Rev. Lett. 85, 4502-4505 (2000).
[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]

Other (3)

D. K. Arrowsmith and C. M. Place, Dynamical Systems: Differential Equations, Maps and Chaotic Behavior (Chapman & Hall, 1992).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, 1988).

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Figures (5)

Fig. 1
Fig. 1

Numerical solution of the system of autonomous equations (54) for the arbitrary initial conditions X 0 and Y 0 . The phase plane ( X , Y ) has the unique singular point (1,0), which is a stable focus. Here all trajectories direct to the point (1,0) when σ 0 > 0 . In the case σ 0 < 0 the trajectories are the same but they have the opposite directions and hence the point (1,0) is an unstable focus. The fixed point (1,0) is globally stable when σ 0 > 0 .

Fig. 2
Fig. 2

Numerical solution of the system (67) [normalized variables Eq. (71)] for case 1 with different initial conditions X 0 , Y 0 and ν = 5 . The phase portraits show that ( X n ( T ) , Y n ( T ) ) ( 1 , 1 ) asymptotically when T and hence ( X ( T ) , Y ( T ) ) ( X a ( T ) , Y a ( T ) ) for arbitrary initial conditions when T .

Fig. 3
Fig. 3

Numerical solution of the system (67) [normalized variables Eq. (71)] for case 2 with different initial conditions X 0 , Y 0 and ν = 3 . The phase portraits show that ( X n ( T ) , Y n ( T ) ) ( 1 , 1 ) asymptotically when T and hence ( X ( T ) , Y ( T ) ) ( X a ( T ) , Y a ( T ) ) for arbitrary initial conditions when T .

Fig. 4
Fig. 4

Relative deviation Δ = ( u an ( s , 0 ) 2 u nu ( s , 0 ) 2 ) u an ( s , 0 ) 2 where u an ( s , 0 ) 2 and u nu ( s , 0 ) 2 are the peak intensity given by analytical expression Eq. (73) and by numerical simulation of Eq. (8), respectively. In case 1, ν = 3 , κ = 1 ; and in case 2, ν = 3 , κ = 1 . We use in numerical simulation initial condition u ( 0 , t ) = exp ( t 2 ) , which yields ε 0 = π 2 .

Fig. 5
Fig. 5

Numerical solution of Eq. (8) with initial condition u ( 0 , t ) = exp ( t 2 ) (unbroken curves) and analytical solution given by Eqs. (73, 74) with ε 0 = π 2 (circles) for case 1, ν = 3 , κ = 1 . We demonstrate the solutions coincide with relative deviation about 0.5% for propagating dimensionless distance s = 26 .

Equations (95)

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

i ψ z = β ( z ) 2 ψ τ τ γ ( z ) ψ 2 ψ + i g ( z ) 2 ψ ,
t = t ( z , τ ) = τ τ c ( z ) τ 0 ( 1 c 0 D ( z ) ) , s = s ( z ) = D ( z ) 2 τ 0 2 ( 1 c 0 D ( z ) ) ,
ψ ( z , τ ) = ρ ( z ) 1 2 τ 0 ( 1 c 0 D ( z ) ) exp ( i Θ ( z , τ ) ) u ( s , t ) ,
Θ ( z , τ ) = a 0 ω 2 4 D ( z ) ω ( τ τ c ( z ) ) + c 0 ( τ τ c ( z ) ) 2 1 c 0 D ( z ) ,
D ( z ) = 2 0 z β ( z ) d z ,
τ c ( z ) = τ c 0 + ω 2 D ( z ) .
ψ ̃ ( z , τ ) = ψ ( z , τ ω 2 D ( z ) ) exp [ i ( ω 2 4 D ( z ) ω τ ) ] .
ψ ̃ ( s ) ( z , τ ) = ψ ( s ) ( z , τ v z ) exp [ i ( v 2 2 β z v β τ ) ] .
i u s = 1 2 m u t t n u 2 u + i 1 2 σ ( s ) u ,
σ ( s ) = 1 α ( z ( s ) ) d d s α ( z ( s ) ) ,
α ( z ) = ( 1 c 0 D ( z ) ) ρ ( z ) exp ( G ( z ) ) .
Σ ( z ) = τ 0 2 ( 1 c 0 D ( z ) ) 2 β ( z ) ( g ( z ) 1 ρ ( z ) d ρ ( z ) d z 2 c 0 β ( z ) 1 c 0 D ( z ) ) .
g ( z ) = 1 ρ ( z ) d ρ ( z ) d z + 2 c 0 β ( z ) 1 c 0 D ( z ) .
A ( s , t ) = f ( s ) F ( T ) ,
ϕ ( s , t ) = φ ( s ) + c ( s ) t 2 ,
T = ε 0 f 2 ( s ) ε ( s ) t ,
and
ε ( s ) = ε 0 exp ( 0 s σ ( s ) d s ) = ε 0 α ( z ( s ) ) α ( 0 ) .
ε ( s ( z ) ) = τ 0 ( 1 c 0 D ( z ) ) ρ ( z ) E ( z ) = ε 0 ρ ( 0 ) ( 1 c 0 D ( z ) ) ρ ( z ) e G ( z ) .
ε 0 = + u ( 0 , t ) 2 d t = τ 0 γ 0 β 0 E 0 ,
E 0 = + ψ ( 0 , τ ) 2 d τ .
1 f 2 d φ d s + ε 2 T 2 ε 0 2 f 6 ( d c d s 2 c 2 ) = F 2 ε 0 2 f 2 2 ε 2 F 1 d 2 F d T 2 ,
d f ( s ) d s = c ( s ) f ( s ) + σ ( s ) 2 f ( s ) .
F 3 ( T ) ε 0 2 f 2 ( s ) 2 ε 2 ( s ) d 2 F ( T ) d T 2 .
A 3 ( s , t ) 1 2 2 A ( s , t ) t 2 .
1 f 2 ( s ) d φ ( s ) d s = F 2 ( T ) ε 2 ( s ) ε 0 2 f 6 ( s ) ( d c ( s ) d s 2 c 2 ( s ) ) T 2 ρ .
F 2 ( T ) = ρ ε 2 ( s ) ε 0 2 f 6 ( s ) ( 2 c 2 ( s ) d c ( s ) d s ) T 2 ,
ε 2 ( s ) ε 0 2 f 6 ( s ) ( 2 c 2 ( s ) d c ( s ) d s ) = b ,
F ( T ) = ( 1 t 2 η 2 ( s ) ) 1 2 ϑ ( η ( s ) t ) ,
η ( s ) 3 ε ( s ) 4 f 2 ( s ) ,
d φ ( s ) d s = f 2 ( s ) ,
d c ( s ) d ( s ) = 2 c 2 ( s ) 16 9 f 6 ( s ) ε 2 ( s ) .
φ ( s ) = φ 0 + 3 4 0 s ε ( s ) η ( s ) d s .
c ( s ) = 1 2 η ( s ) d η ( s ) d s .
2 3 η 2 ( s ) d 2 η ( s ) d s 2 = ε ( s ) ,
ε ( s ) = τ 0 γ 0 E 0 β 0 exp ( 0 s σ ( s ) d s ) .
η ( s ) ε ( s ) 1 .
A ( s , t ) = ( 3 ε ( s ) 4 η ( s ) ) 1 2 ( 1 t 2 η 2 ( s ) ) 1 2 ϑ ( η ( s ) t ) ,
ϕ ( s , t ) = φ 0 + 3 4 0 s ε ( s ) η ( s ) d s 1 2 η ( s ) d η ( s ) d s t 2 .
η ( s ) s = s 0 = η ( s 0 ) ,
d η ( s ) d s s = s 0 = 2 η ( s 0 ) c ( s 0 ) ,
U ( z , τ ) = ( 3 E ( z ) 4 w ( z ) ) 1 2 ( 1 ( τ τ c ( z ) ) 2 w 2 ( z ) ) 1 2 ϑ ( w ( z ) τ τ c ( z ) ) ,
Φ ( z , τ ) = Φ 0 ( z ) ω ( τ τ c ( z ) ) 1 2 β ( z ) w ( z ) d w ( z ) d z ( τ τ c ( z ) ) 2 ,
Φ 0 ( z ) = Φ 0 + 3 4 0 z γ ( z ) w ( z ) E ( z ) d z ω 2 4 D ( z ) .
E ( z ) = E 0 exp ( G ( z ) ) ,
w ( z ) = τ 0 ( 1 c 0 D ( z ) ) η ( s ( z ) ) .
2 3 w 2 ( z ) d d z ( 1 β ( z ) d w ( z ) d z ) = γ ( z ) E ( z ) .
w ( z ) E ( z ) ρ ( z ) 1 1 ,
β ( z ) = β 0 ( 1 + k z ) 1 , γ ( z ) = γ 0 ( 1 + k z ) ν , g ( z ) = g 0 ( 1 + k z ) 1 ,
G ( z ) = ( g 0 k ) ln ( 1 + k z ) ,
α ( z ) = γ 0 β 0 ( 1 + k z ) μ ,
μ = 1 + ν + g 0 k .
D ( z ) = 2 β 0 k ln ( 1 + k z ) ,
s = β 0 k τ 0 2 ln ( 1 + k z ) .
ε ( s ) = ε 0 exp ( σ 0 s ) , σ 0 = k μ τ 0 2 β 0 .
2 3 η 2 ( s ) d 2 η ( s ) d s 2 = ε 0 exp ( σ 0 s ) .
η ( s ) = η 0 exp ( 1 3 σ 0 s ) , η 0 = 3 ( ε 0 2 σ 0 2 ) 1 3 .
w ( z ) = w 0 ( 1 + k z ) μ 3 , w 0 = 3 ( β 0 γ 0 E 0 2 k 2 μ 2 ) 1 3 .
U ( z , τ ) = U 0 ( z ) ( 1 ( τ τ c ( z ) ) 2 w 2 ( z ) ) 1 2 ϑ ( w ( z ) τ τ c ( z ) ) ,
U 0 ( z ) = ( 3 E 0 4 w 0 ) 1 2 ( 1 + k z ) μ 3 ( 1 + ν ) 2 ,
τ c ( z ) = τ c 0 + β 0 ω k ln ( 1 + k z ) .
δ ω ( z , τ ) = Φ τ ( z , τ ) = ω + μ k 3 β 0 ( τ τ c ( z ) ) .
w 0 γ 0 E 0 β 0 ( 1 + k z ) 4 μ 3 1 .
d 2 q d s 2 + 2 σ 0 3 d q d s = d V ( q ) d q ,
V ( q ) = σ 0 2 18 q 2 + 3 ε 0 2 q 1 .
q 0 = 3 ( ε 0 2 σ 0 2 ) 1 3 .
T = σ 0 s , X = q q 0 , Y = d X d T ,
d X d T = Y , d Y d T = 2 3 Y 1 9 ( X 1 X 2 ) .
η ( s ) = 3 ( ε 0 2 σ 0 2 ) 1 3 exp ( 1 3 σ 0 s ) .
1 2 ( d η d s ) 2 + 3 ε 0 2 η = constant .
γ ( z ) = γ 0 β ( z ) β 0 ( 1 c 0 D ( z ) ) exp ( σ 0 D ( z ) 2 τ 0 2 ( 1 c 0 D ( z ) ) G ( z ) ) .
β ( z ) = β 0 γ ( z ) γ 0 ( 1 + c 0 K ( Γ ( z ) ) ) exp ( G ( z ) σ 0 K ( Γ ( z ) ) 2 τ 0 2 ) ,
Γ ( z ) = 2 β 0 γ 0 0 z γ ( z ) e G ( z ) d z ,
R ( x ) = 0 x ( 1 + c 0 t ) 1 exp ( σ 0 t 2 τ 0 2 ) d t ,
β ( z ) = β 0 γ ( z ) γ 0 ( 1 + σ 0 β 0 τ 0 2 γ 0 0 z γ ( z ) e G ( z ) d z ) 1 e G ( z ) .
g ( z ) = 1 ρ ( z ) d ρ ( z ) d z + 2 c 0 β ( z ) 1 c 0 D ( z ) + σ 0 β ( z ) τ 0 2 ( 1 c 0 D ( z ) ) 2 .
w ( z ) = 3 ( γ 0 E 0 τ 0 4 2 β 0 σ 0 2 ) 1 3 ( 1 c 0 D ( z ) ) exp ( σ 0 D ( z ) 6 τ 0 2 ( 1 c 0 D ( z ) ) ) .
η ( s ) = η 0 ( 1 + κ s ) p .
ε ( s ) = ε 0 ( 1 + κ s ) ν ,
p = 1 3 ( ν + 2 ) , η 0 = 3 ( ε 0 2 ( ν 1 ) ( ν + 2 ) κ 2 ) 1 3 .
η 0 ε 0 ( 1 + κ s ) ( 4 ν + 2 ) 3 1 ,
d X d T = Y , d X d T = 3 2 ( 1 ± T ) ν X 2 ,
T = κ s , η ( s ) = ( ε 0 κ 2 ) 1 3 X ( T ) , Y = d X d T .
X a ( T ) = 3 [ 2 ( ν 1 ) ( ν + 2 ) ] 1 3 ( 1 ± T ) ( ν + 2 ) 3 ,
Y a ( T ) = d X a d T = ± ( ν + 2 ) [ 2 ( ν 1 ) ( ν + 2 ) ] 1 3 ( 1 ± T ) ( ν 1 ) 3 .
X n ( T ) = X ( T ) X a ( T ) , Y n ( T ) = Y ( T ) Y a ( T ) .
σ ( s ) = 1 ε ( s ) d ε ( s ) d s = ν κ 1 + κ s .
u ( s , t ) 2 = 3 ε 0 4 η 0 ( 1 + κ s ) 2 ( ν 1 ) 3 ( 1 t 2 η 2 ( s ) ) ϑ ( η ( s ) t ) ,
δ ω ( s , t ) ϕ t ( s , t ) = κ ( ν + 2 ) 3 ( 1 + κ s ) t .
Σ ( z ) = ν κ ( 1 + κ D ( z ) 2 τ 0 2 ( 1 c 0 D ( z ) ) ) 1 ,
w ( z ) = τ 0 η 0 ( 1 c 0 D ( z ) ) ( 1 + κ D ( z ) 2 τ 0 2 ( 1 c 0 D ( z ) ) ) ( ν + 2 ) 3 .
d d T ( X 1 Y 1 ) = [ 0 1 1 3 2 3 ] ( X 1 Y 1 ) , A [ 0 1 1 3 2 3 ] .
d d T ( X 2 Y 2 ) = [ α β β α ] ( X 2 Y 2 ) , B [ α β β α ] .
d R d T = α R , d Φ d T = β ,
R ( T ) = R 0 exp ( α T ) , Φ ( T ) = Φ 0 + β T ,

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