Abstract

We show that under certain conditions the dark-soliton solutions of the defocusing nonlinear Schrödinger equation can be less stable than their bright counterparts. This is due to the tendency of a dark soliton under external perturbation to generate dispersive waves that organize themselves into a shelf around the pulse wings, as exemplified by the simple case of dark solitons under small linear damping or amplification. The shelf generation is a nonadiabatic process: The shelf area is of order unity after a propagation distance that scales as the inverse of the perturbation magnitude. This is in contrast with the bright-soliton dynamics under similar circumstances. We analyze the effect of this extra source of dispersive waves on possible dark-soliton-based long-distance communication systems. We find that the second-order (in the normalized amplifier spacing) perturbation that results from averaging does not lead to shelf formation, whereas a control device in general causes it, unless the controller parameters are chosen according to a simple criterion aimed at suppressing dispersive wave generation. We illustrate the effectiveness of this criterion with extensive numerical simulations.

© 1997 Optical Society of America

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  1. V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971) [Sov. Phys. JETP 34, 62 (1972)].
  2. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
    [CrossRef]
  3. W. Zhao and E. Bourkoff, Opt. Lett. 14, 703 (1989).
    [CrossRef] [PubMed]
  4. M. Lisak, D. Anderson, and B. A. Malomed, Opt. Lett. 16, 1936 (1991).
    [CrossRef] [PubMed]
  5. J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990).
    [CrossRef]
  6. Y. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
    [CrossRef]
  7. J. Gordon and H. Haus, Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  8. J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 16, 1578 (1991).
    [CrossRef] [PubMed]
  9. Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
    [CrossRef]
  10. Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
    [CrossRef]
  11. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
    [CrossRef] [PubMed]
  12. D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
    [CrossRef]
  13. M. Nakazawa and K. Suzuki, Electron. Lett. 31, 1076, 1084 (1995).
  14. Yu. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
    [CrossRef]
  15. V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 48, 252 (1978); D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
    [CrossRef]
  16. D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
    [CrossRef]
  17. V. G. Makhankov, Soliton Phenomenology (Kluwer, Dordrecht, The Netherlands, 1990).
  18. V. V. Konotop and V. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
    [CrossRef]
  19. A. Hasegawa and Y. Kodama, Opt. Lett. 15, 443 (1990);Phys. Rev. Lett. 66, 161 (1991).
    [CrossRef]
  20. A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).
  21. H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
    [CrossRef] [PubMed]
  22. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1994).
  23. L. D. Faddeev and L. A. Takhtadjian, Hamiltonian Approach to Soliton Theory (Springer-Verlag, Berlin, 1986).
  24. I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. E A47, 1582 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  28. H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
    [CrossRef]

1996 (1)

1995 (4)

H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
[CrossRef]

H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

M. Nakazawa and K. Suzuki, Electron. Lett. 31, 1076, 1084 (1995).

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
[CrossRef]

1994 (4)

V. V. Konotop and V. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
[CrossRef]

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
[CrossRef]

Y. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

1993 (2)

Yu. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. E A47, 1582 (1993).
[CrossRef]

1992 (1)

1991 (2)

1990 (2)

J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990).
[CrossRef]

A. Hasegawa and Y. Kodama, Opt. Lett. 15, 443 (1990);Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

1989 (1)

1988 (1)

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

1987 (1)

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

1986 (1)

1978 (1)

V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 48, 252 (1978); D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

1971 (1)

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1994).

Allen, K. M.

K. M. Allen, N. J. Doran, N. J. Smith, and J. A. R. Williams, “Control strategies for temporal dark solitons,” in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 236.

Anderson, D.

Barthelemy, A.

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Bourkoff, E.

Chamberlin, R. P.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
[CrossRef]

Dong, L.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
[CrossRef]

Doran, N. J.

K. M. Allen, N. J. Doran, N. J. Smith, and J. A. R. Williams, “Control strategies for temporal dark solitons,” in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 236.

Emplit, Ph.

Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
[CrossRef]

J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 16, 1578 (1991).
[CrossRef] [PubMed]

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Evangelides, S. G.

Faddeev, L. D.

L. D. Faddeev and L. A. Takhtadjian, Hamiltonian Approach to Soliton Theory (Springer-Verlag, Berlin, 1986).

Froehly, C.

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Gerdjikov, V. S.

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. E A47, 1582 (1993).
[CrossRef]

Giannini, J. A.

J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990).
[CrossRef]

Goedde, C. G.

Gordon, J.

Gordon, J. P.

Haeletrman, M.

Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
[CrossRef]

J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 16, 1578 (1991).
[CrossRef] [PubMed]

Hamaide, J. P.

Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
[CrossRef]

J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 16, 1578 (1991).
[CrossRef] [PubMed]

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Hasegawa, A.

H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
[CrossRef]

A. Hasegawa and Y. Kodama, Opt. Lett. 15, 443 (1990);Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).

Haus, H.

Hawkins, R. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Ikeda, H.

H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
[CrossRef]

Joseph, R. I.

J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990).
[CrossRef]

Karpman, V. I.

V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 48, 252 (1978); D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

Kath, W. L.

Kim, A. D.

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Kivshar, Y. S.

Y. S. Kivshar, J. P. Hamaide, Ph. Emplit, and M. Haeletrman, Opt. Lett. 19, 9 (1994).
[CrossRef]

Y. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

Kivshar, Yu.

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
[CrossRef]

Yu. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

Kodama, Y.

A. Hasegawa and Y. Kodama, Opt. Lett. 15, 443 (1990);Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).

Konotop, V. V.

V. V. Konotop and V. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Leaird, D. E.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Lisak, M.

Makhankov, V. G.

V. G. Makhankov, Soliton Phenomenology (Kluwer, Dordrecht, The Netherlands, 1990).

Malomed, B. A.

Maslov, E. M.

V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 48, 252 (1978); D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

Matsumoto, M.

H. Ikeda, M. Matsumoto, and A. Hasegawa, Opt. Lett. 20, 1113 (1995).
[CrossRef] [PubMed]

H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
[CrossRef]

Mollenauer, L. F.

Nakazawa, M.

M. Nakazawa and K. Suzuki, Electron. Lett. 31, 1076, 1084 (1995).

Payne, D. N.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
[CrossRef]

Reynaud, F.

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Richardson, D. J.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (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)].

Smith, N. J.

K. M. Allen, N. J. Doran, N. J. Smith, and J. A. R. Williams, “Control strategies for temporal dark solitons,” in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 236.

Stepanyants, Yu. A.

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
[CrossRef]

Suzuki, K.

M. Nakazawa and K. Suzuki, Electron. Lett. 31, 1076, 1084 (1995).

Takhtadjian, L. A.

L. D. Faddeev and L. A. Takhtadjian, Hamiltonian Approach to Soliton Theory (Springer-Verlag, Berlin, 1986).

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Tomlinson, W. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Uzunov, I. M.

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. E A47, 1582 (1993).
[CrossRef]

Vekslerchik, V.

V. V. Konotop and V. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Weiner, A. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Williams, J. A. R.

K. M. Allen, N. J. Doran, N. J. Smith, and J. A. R. Williams, “Control strategies for temporal dark solitons,” in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 236.

Yang, X.

Y. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[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.

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

Electron Lett. (1)

H. Ikeda, M. Matsumoto, and A. Hasegawa, Electron Lett. 31, 482 (1995).
[CrossRef]

Electron. Lett. (2)

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, Electron. Lett. 30, 1326 (1994).
[CrossRef]

M. Nakazawa and K. Suzuki, Electron. Lett. 31, 1076, 1084 (1995).

IEEE J. Quantum Electron. (2)

Yu. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).
[CrossRef]

J. A. Giannini and R. I. Joseph, IEEE J. Quantum Electron. 26, 2109 (1990).
[CrossRef]

Opt. Commun. (1)

Ph. Emplit, J. P. Hamaide, F. Reynaud, C. Froehly, and A. Barthelemy, Opt. Commun. 62, 374 (1987).
[CrossRef]

Opt. Lett. (9)

Phys. Rev. E (4)

I. M. Uzunov and V. S. Gerdjikov, Phys. Rev. E A47, 1582 (1993).
[CrossRef]

Y. S. Kivshar and X. Yang, Phys. Rev. E 49, 1657 (1994).
[CrossRef]

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. Kivshar, Phys. Rev. E 51, 5016 (1995).
[CrossRef]

V. V. Konotop and V. Vekslerchik, Phys. Rev. E 49, 2397 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, and W. J. Tomlinson, Phys. Rev. Lett. 61, 2445 (1988).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. I. Karpman and E. M. Maslov, Sov. Phys. JETP 48, 252 (1978); D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

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

Other (5)

V. G. Makhankov, Soliton Phenomenology (Kluwer, Dordrecht, The Netherlands, 1990).

K. M. Allen, N. J. Doran, N. J. Smith, and J. A. R. Williams, “Control strategies for temporal dark solitons,” in Nonlinear Guided Waves and Their Applications, Vol. 6 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), p. 236.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1994).

L. D. Faddeev and L. A. Takhtadjian, Hamiltonian Approach to Soliton Theory (Springer-Verlag, Berlin, 1986).

A. Hasegawa and Y. Kodama, Solitons in Optical Communication (Oxford U. Press, Oxford, 1995).

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Figures (5)

Fig. 1
Fig. 1

Black-soliton evolution under small damping, γ =-0.1.

Fig. 2
Fig. 2

Black-soliton evolution under small amplification, γ =0.1.

Fig. 3
Fig. 3

Black-soliton evolution under δ=-0.19, γ1=0.39, and γ2=-0.2.

Fig. 4
Fig. 4

Summary of the results of numerical simulation of Eq. (19) in domain D.

Fig. 5
Fig. 5

Phase profile of the solution of Eq. (19): solid line, initial condition; dashed curve, at z=20 for γ1=0.4, γ2 =-0.22, and δ=-0.18, which corresponds to a point off to the right of the no-shelf line in Fig. 4; dotted curve (which coincides almost completely with the solid line), at z=20 for γ1=0.4, γ2=-0.25, and δ=-0.15, which corresponds to a point on the no-shelf line in Fig. 4.

Equations (28)

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

irz-rtt/2+r2r*=0,
rs=exp(iu02z)[λ-iν tanh(νθ)],
θ=t+λz
u02=ν2+λ2
irz-rtt/2+r2r*=iγr,
r(z, t)=u0(z)exp[iβ(z)]
du0dz=γu0,dβdz=u02.
I1=- dt(u02-rr*),
I2=i2- dt(rrt*-r*rt)-u02 arg(r)|t=-t=+,
I3=- dt[rtrt*+(u02-rr*)2].
I1=2ν,I2=-2νλ+2u02 sin-1(ν/u0),
I3=8ν3/3.
dI3dz=2γI3+2γ - dt(u02-rr*)2.
dνdz=γνν=ν0eγz.
dI1dz=2γI1.
dνdz=2γνν=ν0 exp(2γz),
irz-rtt/2+r2r*=iγr+iG(z)r+iR,
G(z)=G0 n=0N δ(z-nza),
ivz+σvtt/2+Iv2v*=za2P+O(za3).
P=-σα(2v2v*2vtt+2v3v*vtt*+vv*2vt2+v3vt*2+6v2v*vtvt*),
ivz-vtt/2+v2v*=i(δv+γ1v2v*+γ2v3v*2),
du0dz=δu0+γ1u03+γ2u05,
δ+γ1+γ2=0,
γ1+2γ2<0.
dνdz=δν+γ1(5u02ν/3-2ν3/3)+γ2(7u04ν/3-28u02ν3/15+8ν5/15).
5γ1+6γ2>0.
dνdz=2δν+4γ1(u02ν-ν3/3)+γ2(6u04ν-4u02ν3+16ν5/15),
5γ1+8γ2=0,

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