Abstract

We consider the analytically asymptotic evolution of a pulse governed by the nonlinear Schrödinger equation with an additional term that accounts for the Raman intrapulse scattering. This term is taken in the simplest quasi-instantaneous approximation. First we find an asymptotic analytical solution describing a tail generated by action of the Raman term. An important result is that this solution always remains essentially nonlinear, which drastically changes its structure compared with the known linear approximations based on the Airy functions. It is shown that a minimum temporal scale of the tail increases with the propagation distance, so that the quasi-instantaneous approximation for the Raman term remains valid. Our solution may directly apply to the description of the tails discovered in recent numerical and laboratory experiments. Next, we consider a late asymptotic stage of evolution of a Raman-driven localized pulse. We demonstrate that, beyond the well-known approximation in which the pulse (soliton) is regarded as a mechanical particle driven by a constant force, the pulse propagating in a long lossless fiber will slowly decay into the oscillating tail generated behind it. Using the analytical solution for the tail, the conservation of energy, and some natural assumptions, we predict that, asymptotically, the peak power of the soliton will vanish inversely proportionally to the traveled distance. A real optical fiber should allow for the predicted process if it is much longer than several soliton periods, which may be realistic for the ultrashort solitons. Finally we demonstrate that, in the model incorporating gain and losses, the tail can trigger a long-term instability of the solitary pulse, which has been observed in numerical simulations.

© 1995 Optical Society of America

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References

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  1. A. Hasegawa, Solitons in Optical Fibers (Springer-Verlag, Berlin, 1989); G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
    [CrossRef]
  2. V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].
  3. D. J. Kaup, SIAM J. Applied Mat. 31, 121 (1976).
    [CrossRef]
  4. J. P. Gordon, Opt. Lett. 11, 662 (1986); F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
    [CrossRef] [PubMed]
  5. M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
    [CrossRef]
  6. E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
    [CrossRef]
  7. K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
    [CrossRef]
  8. Y. Kodama and K. Nozaki, Opt. Lett. 12, 1038 (1987); S. Chi and S. Wen, Opt. Lett. 14, 1216 (1989).
    [CrossRef] [PubMed]
  9. E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
    [CrossRef] [PubMed]
  10. C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
    [CrossRef] [PubMed]
  11. L. Gagnon and P. A. Bélanger, Opt. Lett. 15, 466 (1990).
    [CrossRef] [PubMed]
  12. J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
    [CrossRef] [PubMed]
  13. B. A. Malomed, Phys. Scr. 38, 66 (1988).
    [CrossRef]
  14. D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
    [CrossRef] [PubMed]
  15. A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
    [CrossRef]
  16. A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
    [CrossRef]
  17. Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
    [CrossRef] [PubMed]
  18. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
    [CrossRef] [PubMed]
  19. Y. H. Ichikawa, Phys. Scr. 20, 296 (1979).
    [CrossRef]

Anderson, D.

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Bar-Joseph, I.

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Bélanger, P. A.

L. Gagnon and P. A. Bélanger, Opt. Lett. 15, 466 (1990).
[CrossRef] [PubMed]

Blow, K. J.

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

Bourkoff, E.

E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
[CrossRef] [PubMed]

Brodin, G.

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Chemla, D. S.

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Chen, C.-J.

C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
[CrossRef] [PubMed]

Christodoulides, D. N.

E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
[CrossRef] [PubMed]

Dianov, E. M.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Doran, N. J.

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

Evangelides, S. G.

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

Gagnon, L.

L. Gagnon and P. A. Bélanger, Opt. Lett. 15, 466 (1990).
[CrossRef] [PubMed]

Gordon, J. P.

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

J. P. Gordon, Opt. Lett. 11, 662 (1986); F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef] [PubMed]

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Grudinin, A. B.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Solitons in Optical Fibers (Springer-Verlag, Berlin, 1989); G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
[CrossRef]

Ichikawa, Y. H.

Y. H. Ichikawa, Phys. Scr. 20, 296 (1979).
[CrossRef]

Islam, M. N.

C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
[CrossRef] [PubMed]

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Joseph, R. J.

E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
[CrossRef] [PubMed]

Kaup, D. J.

D. J. Kaup, SIAM J. Applied Mat. 31, 121 (1976).
[CrossRef]

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Khaidarov, D. V.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Kodama, Y.

Y. Kodama and K. Nozaki, Opt. Lett. 12, 1038 (1987); S. Chi and S. Wen, Opt. Lett. 14, 1216 (1989).
[CrossRef] [PubMed]

Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
[CrossRef] [PubMed]

Korobkin, D. V.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Lisak, M.

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Lucek, J. K.

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

Malomed, B. A.

B. A. Malomed, Phys. Scr. 38, 66 (1988).
[CrossRef]

Menyuk, C. R.

C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

Nozaki, K.

Y. Kodama and K. Nozaki, Opt. Lett. 12, 1038 (1987); S. Chi and S. Wen, Opt. Lett. 14, 1216 (1989).
[CrossRef] [PubMed]

Prokhorov, A. M.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Sergeev, A.

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

Sergeev, A. M.

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Serkin, V. N.

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Stenflo, L.

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

Stolen, R. H.

C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
[CrossRef] [PubMed]

Sucha, G.

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Vanin, E.

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

Vanin, E. V.

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Wabnitz, S.

Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
[CrossRef] [PubMed]

Wegener, M.

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

Wood, D.

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

Zhao, W.

E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
[CrossRef] [PubMed]

Other (19)

A. Hasegawa, Solitons in Optical Fibers (Springer-Verlag, Berlin, 1989); G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
[CrossRef]

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].

D. J. Kaup, SIAM J. Applied Mat. 31, 121 (1976).
[CrossRef]

J. P. Gordon, Opt. Lett. 11, 662 (1986); F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[CrossRef] [PubMed]

M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, J. Opt. Soc. Am. B 6, 1149 (1989).
[CrossRef]

E. M. Dianov, A. B. Grudinin, D. V. Khaidarov, D. V. Korobkin, A. M. Prokhorov, and V. N. Serkin, Fiber Integr. Opt 8, 61 (1989); J. Schütz, W. Hodel, and H. P. Weber, Opt. Commun. 107, 170 (1994).
[CrossRef]

K. J. Blow, N. J. Doran, and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988).
[CrossRef]

Y. Kodama and K. Nozaki, Opt. Lett. 12, 1038 (1987); S. Chi and S. Wen, Opt. Lett. 14, 1216 (1989).
[CrossRef] [PubMed]

E. Bourkoff, W. Zhao, D. N. Christodoulides, and R. J. Joseph, Opt. Lett. 12, 272 (1987); W. Hodel and H. P. Weber, Opt. Lett. 12, 924 (1987); G. P. Agrawal, Opt. Lett. 15, 224 (1990); 16, 226 (1991); B. J. Hong and C. C. Yang, J. Opt. Soc. Am. B 8, 1114 (1991); K. C. Chan and H. F. Liu, Opt. Lett. 18, 1150 (1993); A. Höök and M. Karlsson, Opt. Lett. 18, 1388 (1993); S. Liu and W. Wang, Opt. Lett. 15, 1911 (1993).
[CrossRef] [PubMed]

C.-J. Chen, C. R. Menyuk, M. N. Islam, and R. H. Stolen, Opt. Lett. 16, 1647 (1991); S. Wabnitz, Opt. Lett. 17, 923 (1992); R. J. Hawkins and C. R. Menyuk, Opt. Lett. 18, 1999 (1993).
[CrossRef] [PubMed]

L. Gagnon and P. A. Bélanger, Opt. Lett. 15, 466 (1990).
[CrossRef] [PubMed]

J. K. Lucek and K. J. Blow, Phys. Rev. A 45, 6666 (1992).
[CrossRef] [PubMed]

B. A. Malomed, Phys. Scr. 38, 66 (1988).
[CrossRef]

D. J. Kaup, Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

A. Sergeev, E. Vanin, D. Anderson, M. Lisak, and L. Stenflo, Opt. Commun. 101, 219 (1993).
[CrossRef]

A. M. Sergeev, E. V. Vanin, G. Brodin, L. Stenflo, D. Anderson, and M. Lisak, Phys. Lett. A 177, 130 (1993).
[CrossRef]

Y. Kodama and S. Wabnitz, Opt. Lett. 19, 162 (1994).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

Y. H. Ichikawa, Phys. Scr. 20, 296 (1979).
[CrossRef]

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Equations (41)

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

i u z + ½ u τ τ + u 2 u = ( u 2 ) τ u .
u sol = η sech { η [ τ - T sol ( z ) ] } exp [ - i ω sol τ + i ϕ sol ( z ) ] ,
d d z T sol = - ω sol ,             d d z ϕ sol = ½ ( ω sol 2 - η 2 ) .
d d z ω sol = - 8 15 η 4 .
W = - + u ( τ ) 2 d τ ,
W sol = 2 η .
u tail ( z , τ ) = a [ τ - T tail ( z ) ] exp { - i ω tail ( z ) [ τ - T tail ( z ) ] + i ϕ ( z ) } ,
T ˙ tail = - ω tail ,             ϕ ˙ = ½ ω 2 ,
½ a + ω ˙ tail s a - 2 a 2 a + a 3 = 0 ,
a ( s ) = A 1 ( z ) cos ψ ( s ) + A 3 ( s ) cos [ 3 ψ ( s ) ] + B 3 ( s ) sin [ 3 ψ ( s ) ] + ,
( ψ ) 2 = 2 ω ˙ tail s + 3 2 A 1 2 - 3 A 1 A 1 + A 1 A 1 ,
A 1 + 1 2 A 1 ψ ψ - 1 2 A 0 3 = 0 ,
A 3 = A 1 3 - 2 A 1 A 1 2 16 ( ψ ) 2 ,
B 3 = A 1 3 8 ψ .
ψ = ( - 2 ω ˙ tail ) 1 / 2 ( - s ) 3 / 2 + .
A 1 = 1 v ( C + 2 v 2 ) 1 / 2 ,
s 2 ( - ω ˙ tail ) 1.
ψ 1.
W tail = 1 4 ln [ 4 2 S ( C + 2 s 0 1 / 2 ) 2 ] .
s 0 ~ ω ˙ tail - 1 / 3 .
C ~ s 0 1 / 2 .
s 0 [ η ( z ) ] 2 ~ 1.
ω ˙ tail 1 / 3 ~ η 2 .
( η ) 2 1.
W tail = ( 4 ) - 1 ln ( η 2 T sol ) .
η 0 = η + ( 4 ) - 1 ln ( η 2 T sol ) .
T ¨ sol = 8 15 η 4 .
T sol = V z - 8 15 ( V 2 ) - 1 ln ( z / z 0 )
η 2 ( z ) = ( V ) - 1 z - 1 .
T 0 ~ ( η 0 2 ) - 1 .
T 0 4 15 η 0 4 z 0 2
z 0 ~ ( η 0 3 ) - 1 .
T sol η 0 z - 8 15 ( η 0 2 ) - 1 ln ( η 0 3 z ) ,
η 2 ( z ) ( η 0 ) - 1 z - 1 .
λ min = 2 π ( 2 s ω ˙ tail ) - 1 / 2 ,
( δ τ ) min ~ 2 π η 0 z .
η ~ 0.05 ,
z 0 / τ sol ( 2 π η 0 ) - 1 .
a ( σ ) = θ ( - σ ) σ - 1 / 2 cos ( 2 3 σ 3 / 2 ) ,
a ^ ( Ω ) = - + exp ( i Ω σ ) a ( σ ) d σ ,
i u z + ½ u τ τ + u 2 u = ( u 2 ) τ u + i γ 0 u + i γ 1 u τ τ ,

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