Abstract

We present a new asymptotically exact analytical similariton solution of the generalized nonlinear Schrdinger equation for pulses propagating in fiber amplifiers and lasers with normal dispersion including the effect of gain saturation. Numerical simulations are in excellent agreement with this analytical solution describing self-similar linearly chirped parabolic pulses. We have also found that for small enough values of the dimensionless saturation energy parameter the fiber amplifiers and lasers can generate a new type of linearly chirped self-similar pulses, which we call Hyper-Gaussian similaritons. The analytical Hyper-Gaussian similariton solution of the generalized nonlinear Schrdinger equation is also in a good agreement with numerical simulations.

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  1. C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
    [CrossRef] [PubMed]
  2. A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
    [CrossRef]
  3. V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
    [CrossRef]
  4. T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. 23, 268–270 (1998).
    [CrossRef]
  5. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
    [CrossRef] [PubMed]
  6. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).
    [CrossRef]
  7. M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]
  8. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000).
    [CrossRef]
  9. 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]
  10. V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006).
    [CrossRef]
  11. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
    [CrossRef]
  12. A. Ruehl, A. Marcinkevicius, M. E. Fermann, and I. Hartl, “80 W, 120 fs Yb-fiber frequency comb,” Opt. Lett. 35, 3015–3017 (2010).
    [CrossRef] [PubMed]
  13. F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003).
    [CrossRef] [PubMed]
  14. F. O. 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]
  15. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, 2001).
  16. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
    [CrossRef]
  17. B. G. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. 35, 2466–2468 (2010).
    [CrossRef] [PubMed]
  18. V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. 81, 023815 (2010).
    [CrossRef]

2010 (4)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
[CrossRef]

V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. 81, 023815 (2010).
[CrossRef]

B. G. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. 35, 2466–2468 (2010).
[CrossRef] [PubMed]

A. Ruehl, A. Marcinkevicius, M. E. Fermann, and I. Hartl, “80 W, 120 fs Yb-fiber frequency comb,” Opt. Lett. 35, 3015–3017 (2010).
[CrossRef] [PubMed]

2007 (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

2006 (1)

2004 (1)

F. O. 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)

2002 (1)

2000 (3)

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

M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

1998 (1)

1993 (1)

1992 (2)

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

1991 (1)

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Afanas’ev, A. A.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Agrawal, G. P.

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

Anderson, D.

Bale, B. G.

Bergman, K.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Buckley, J. R.

F. O. 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]

F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003).
[CrossRef] [PubMed]

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
[CrossRef]

Clark, W. G.

F. O. 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]

F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003).
[CrossRef] [PubMed]

de Sterke, C. M.

Desaix, M.

Dudley, J. M.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[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]

M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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 fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000).
[CrossRef]

Fermann, M. E.

A. Ruehl, A. Marcinkevicius, M. E. Fermann, and I. Hartl, “80 W, 120 fs Yb-fiber frequency comb,” Opt. Lett. 35, 3015–3017 (2010).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Hartl, I.

Harvey, J. D.

Ilday, F. O.

F. O. 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]

F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003).
[CrossRef] [PubMed]

Jakyte, R.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Karlsson, M.

Kruglov, V. I.

V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. 81, 023815 (2010).
[CrossRef]

V. I. Kruglov and J. D. Harvey, “Asymptotically exact parabolic solutions of the generalized nonlinear Schrodinger equation with varying parameters,” J. Opt. Soc. Am. B 23, 2541–2550 (2006).
[CrossRef]

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. Thomson, 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 fiber amplifiers,” Opt. Lett. 25, 1753–1755 (2000).
[CrossRef]

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Krylov, D.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Levi, D.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Lim, H.

Lisak, M.

Logvin, Yu. A.

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

Marcinkevicius, A.

Mechin, D.

V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. 81, 023815 (2010).
[CrossRef]

Menyuk, C. R.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Millar, P. D.

Millot, G.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Monro, T. M.

Peacock, A. C.

Poladian, L.

Quiroga-Teixeiro, M. L.

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
[CrossRef]

Richardson, D. J.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Ruehl, A.

Samson, B. A.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Sears, S.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Segev, M.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Soljacic, M.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Thomson, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

Volkov, V. M.

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Wabnitz, S.

Winternitz, P.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
[CrossRef]

F. O. 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]

F. O. Ilday, J. R. Buckley, H. Lim, F. W. Wise, and W. G. Clark, “Generation of 50-fs, 5-nj pulses at 1.03 μm from a wave-breaking-free fiber laser,” Opt. Lett. 28, 1365–1367 (2003).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

V. I. Kruglov, Yu. A. Logvin, and V. M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[CrossRef]

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

Nat. Phys. (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A. (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A. 82, 021805 (2010).
[CrossRef]

V. I. Kruglov, D. Mechin, and J. D. Harvey, “All-fiber ring Raman laser generating parabolic pulses,” Phys. Rev. A. 81, 023815 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

F. O. 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]

M. E. Fermann, V. I. Kruglov, B. C. Thomson, 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]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) solutions for distance parameter ξ = 400 with saturation energy parameter ηs = 2 and input energy parameter η0 = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).

Fig. 2
Fig. 2

Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) solutions for distance parameter ξ = 4000 with saturation energy parameter ηs = 2 and input energy parameter η0 = 0.02. The dimensionless energy η (ξ) is demonstrated in the inset diagram (a).

Fig. 3
Fig. 3

Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) pulse for distance parameter ξ = 100 with saturation energy parameter ηs = 100 and input energy parameter η0 = 10−3. The dimensionless energy η (ξ) is illustrated in the inset diagram (a).

Fig. 4
Fig. 4

Pulse power (a) and chirp (b) of numerical (solid line) and analytical (dotted line) pulse for distance parameter ξ = 600 with saturation energy parameter ηs = 100 and input energy parametyer η0 = 10−3. The dimensionless pulse energy η (ξ) is illustrated in the inset diagram (a).

Fig. 5
Fig. 5

Pulse power of numerical HG pulse (solid line), analytical parabolic solution (blue dotted line) and analytical HG pulse power profile (green dotted line) for distance parameters ξ = 8000 (a) and ξ = 14000 (b) with saturation energy parameter ηs = 0.1 and input energy parameter η0 = 10−4. The Hyper-Gaussian pulse power profile is given by Eq. (44).

Fig. 6
Fig. 6

(a) Pulse spectrum of input pulse (dotted line) and numerical output HG pulse (solid line) for distance parameter ξ = 14000. (b) Chirp of numerical output HG pulse (red curve) and analytical HG pulse (blue line) with distance parameter ξ = 14000. Here the saturation energy parameter and input energy parameter are ηs = 0.1 and η0 = 10−4.

Fig. 7
Fig. 7

Power profiles for numerical HG (solid line) and analytical HG (dotted line) pulses for distance parameter ξ = 8000 with four different input pulses and ηs = 0.1, η0 = 10−4. The parabolic analytical solution is also shown (dashed line) which differ substantially from numerical and analytical HG similaritons.

Equations (45)

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i ψ z = β 2 2 ψ τ τ γ | ψ | 2 ψ + i g ( z ) 2 ψ + i g ( z ) 2 σ ψ τ τ ,
g ( z ) = g 0 ( 1 + 1 E s + | ψ ( z , τ ) | 2 d τ ) 1 .
ψ ( z , τ ) = exp ( 1 2 G ( z ) ) ψ ˜ ( z , τ ) , G ( z ) = 0 z g ( z ) d z ,
A ( z , τ ) = E ( z ) E 0 B ( z , τ ) , E ( z ) = E 0 exp ( G ( z ) ) ,
i ψ ˜ z = β 2 2 ψ ˜ τ τ Γ ( z ) | ψ ˜ | 2 ψ ˜ , Γ ( z ) = γ exp ( G ( z ) ) .
2 B B z = β 2 Φ τ τ B 2 + 2 β 2 Φ τ B B τ
Φ z = Γ B 2 + β 2 2 ( Φ τ ) 2 β 2 2 ( B τ τ B ) .
2 γ A 2 β 2 | A τ τ A | ,
𝒡 z = β 2 Φ τ τ 𝒡 + β 2 Φ τ 𝒡 τ
Φ z = Γ 𝒡 + β 2 2 ( Φ τ ) 2 .
A ( z , τ ) = P ( z ) 1 / 2 ( 1 τ 2 τ p ( z ) 2 ) 1 / 2 θ ( τ p ( z ) | τ | ) ,
Φ ( z , τ ) = ϕ 0 + 3 γ 4 0 z E ( z ) τ p ( z ) d z + C ( z ) τ 2 ,
P ( z ) = 3 E ( z ) 4 τ p ( z ) , C ( z ) = 1 2 β 2 τ p ( z ) d τ p ( z ) d z .
d 2 τ p ( z ) d z 2 = ( 3 γ β 2 2 ) E ( z ) τ p ( z ) 2 ,
τ p ( z ) = 3 ( γ β 2 E 0 2 g 2 ) 1 / 3 e g z / 3 .
( 3 γ 2 β 2 ) E ( z ) τ p ( z ) ( 1 τ 2 τ p ( z ) 2 ) 3 .
d E ( z ) d z = g 0 E ( z ) 1 + E ( z ) / E s .
E ( z ) = E 0 + E s g 0 z E s ln E ( z ) E 0 ,
ε ( ξ ) = α 1 + ξ ln ( α ξ ) , α = E s / E 0 ,
τ p ( z ) = ( 3 γ β 2 E s 2 g 0 2 ) 1 / 3 T ( ξ ) ,
d 2 T d ξ 2 = ε T 2 , d ε d ξ = ε 1 + ε .
d 2 W d ε 2 + 1 ε ( 1 + ε ) d W d ε = ( 1 + ε ) 2 ε W 2 .
d 2 W d ε 2 = ε W 2 .
d 2 Y d x 2 + d Y d x = 1 Y 2 .
U d 2 U d x 2 2 3 ( d U d x ) 2 + U d U d x 3 U = 0.
U ( x ) = n m B n m ( ln x ) m x n ,
U ( x ) = 3 x + 5 + 2 ln x + 4 ln x 3 x 31 9 x 2 + 8 ln x 9 x 2 4 ( ln x ) 2 9 x 2 766 81 x 3 + 140 ln x 27 x 3 8 ( ln x ) 2 9 x 3 + 16 ( ln x ) 3 81 x 3 + .
W ( ε ) = ε [ 3 ( σ + ln ε ) + 5 + 2 ln ( σ + ln ε ) + 4 ln ( σ + ln ε ) 3 ( σ + ln ε ) 31 9 ( σ + ln ε ) 2 + 8 ln ( σ + ln ε ) 9 ( σ + ln ε ) 2 4 ( ln ( σ + ln ε ) ) 2 9 ( σ + ln ε ) 2 + ] 1 / 3 .
T ( ξ ) = ( α 1 + ξ ln ( α ξ ) ) [ κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) ] 1 / 3 .
τ p ( z ) = ( 3 γ β 2 E s 2 g 0 2 ) 1 / 3 ( α 1 + g 0 z ln ( α g 0 z ) ) [ κ + 3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) ] 1 / 3 .
P ( z ) = 3 4 ( 2 g 0 2 E s 2 3 β 2 γ ) 1 / 3 [ κ + 3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) ] 1 / 3 ,
C ( z ) = ( g 0 z 1 ) 2 β 2 z ( α 1 + g 0 z ln ( α g 0 z ) ) { 1 + 1 κ + 3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) } .
3 ln ( α 1 + g 0 z ln ( α g 0 z ) ) | κ | .
χ ( ξ , ζ ) = γ g 0 ψ ( z , τ ) , ξ = g 0 z , ζ = g 0 β 2 τ .
i χ ξ = 1 2 χ ζ ζ | χ | 2 χ + i 2 ( 1 + η s 1 + | χ ( ξ , ζ ) | 2 d ζ ) 1 χ .
η s = γ E s g 0 β 2 , η 0 = γ E 0 g 0 β 2 ,
u ( ξ , ζ ) = p ( ξ ) 1 / 2 ( 1 ζ 2 τ ( ξ ) 2 ) 1 / 2 θ ( τ ( ξ ) | ζ | ) ,
τ ( ξ ) = ( 3 η s 2 ) 1 / 3 ( α 1 + ξ ln ( α ξ ) ) [ κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) ] 1 / 3 ,
p ( ξ ) = 3 4 ( 2 η s 2 3 ) 1 / 3 [ κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) ] 1 / 3 .
ω ( ξ , ζ ) = ( ξ 1 ) ξ ( α 1 + ξ ln ( α ξ ) ) { 1 + 1 κ + 3 ln ( α 1 + ξ ln ( α ξ ) ) } ζ .
P ( z , τ ) = Λ E ( z ) w ( z ) exp [ n = 1 σ n ( τ w ( z ) ) n ] ,
w ( z ) = ρ ( γ β 2 E s g 0 ) 1 / 3 ( z c + z ) .
Φ ( z , τ ) = ϕ 0 + Λ γ 0 z E ( z ) μ ( z c + z ) d z τ 2 2 β 2 ( z c + z ) .
Ω ( z , τ ) = Φ τ ( z , τ ) = τ β 2 z .
P ( z , τ ) = Λ E ( z ) w ( z ) exp [ ( τ w ( z ) ) 2 σ ( τ w ( z ) ) 4 ] .

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