Abstract

The propagation of dispersion-managed vector solitons in optical fibers with periodic and random birefringence is studied. With the help of a variational approach, the equations that describe the evolution of pulse parameters are derived. Numerical modeling is performed for variational equations and for fully coupled periodic and stochastic nonlinear Schrödinger equations. It is shown that variational equations can be effectively used to describe the averaged dynamics of dispersion-managed vector solitons with stochastic perturbations. It is shown, analytically and numerically, that dispersion-managed (DM) solitons have the same resistance to random birefringence as do ordinary solitons. The dependence of the mean decay length of a DM vector soliton on the strength of random birefringence and on the energy of the initial pulse is found.

© 2000 Optical Society of America

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References

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  1. N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
    [CrossRef]
  2. E. A. Golovchenko, J. M. Jacob, A. N. Pilipetskii, C. R. Menyuk, and G. M. Carter, “Dispersion-managed solitons in a fiber loop with in-line filtering,” Opt. Lett. 22, 289–291 (1997).
    [CrossRef] [PubMed]
  3. I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
    [CrossRef]
  4. T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
    [CrossRef]
  5. S. Turytsin, “Stability of an optical soliton with Gaussian tails,” Phys. Rev. E 56, R3784–R3787 (1997).
    [CrossRef]
  6. T. Georges, “Soliton interaction in dispersion-managed links,” J. Opt. Soc. Am. B 15, 1553–1560 (1998).
    [CrossRef]
  7. A. H. Liang, H. Toda, and A. Hasegawa, “High-speed soliton transmission in dense periodic fibers,” Opt. Lett. 24, 799–801 (1999).
    [CrossRef]
  8. S. K. Turitsyn, M. P. Fedoruk, and A. Gornakova, “Reduced power optical solitons in fiber lines with short-scale dispersion-management,” Opt. Lett. 24, 869–871 (1999).
    [CrossRef]
  9. B. A. Malomed and N. F. Smyth, “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994).
    [CrossRef]
  10. Yu. S. Kivshar and V. V. Konotop, “Vector solitons in optical fiber with random birefringence,” Kvant. Elektron. (Moscow) 17, 1599–1602 (1990).
  11. T. Ueda and W. L. Kath, “Stochastic simulation of pulses in nonlinear-optical fiber with random birefringence,” J. Opt. Soc. Am. B 11, 818–825 (1991).
    [CrossRef]
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    [CrossRef]
  13. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, Oxford, 1995).
  14. T. Lakoba and D. J. Kaup, “Perturbation theory for Manakov system and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
    [CrossRef]
  15. F. Kh. Abdullaev, J. Bronski, and G. Papanicolaou, “Soliton perturbations and random Kepler problem,” submitted to Physica D.
  16. F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fiber,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef] [PubMed]
  22. D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
    [CrossRef]
  23. T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
    [CrossRef]
  24. J. Gukkenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).
  25. J. N. Kutz, P. Holmes, S. G. Evangelides, Jr., and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  26. X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
    [CrossRef]
  27. N. F. Smyth and A. H. Pincombe, “Effect of radiative loss on pulses in periodically inhomogeneous birefringent optical fibers,” Phys. Rev. E 57, 7231–7238 (1998).
    [CrossRef]

1999 (2)

1998 (6)

T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

T. Georges, “Soliton interaction in dispersion-managed links,” J. Opt. Soc. Am. B 15, 1553–1560 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, Jr., and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

N. F. Smyth and A. H. Pincombe, “Effect of radiative loss on pulses in periodically inhomogeneous birefringent optical fibers,” Phys. Rev. E 57, 7231–7238 (1998).
[CrossRef]

1997 (5)

T. Lakoba and D. J. Kaup, “Perturbation theory for Manakov system and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
[CrossRef]

S. Turytsin, “Stability of an optical soliton with Gaussian tails,” Phys. Rev. E 56, R3784–R3787 (1997).
[CrossRef]

E. A. Golovchenko, J. M. Jacob, A. N. Pilipetskii, C. R. Menyuk, and G. M. Carter, “Dispersion-managed solitons in a fiber loop with in-line filtering,” Opt. Lett. 22, 289–291 (1997).
[CrossRef] [PubMed]

I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

1996 (2)

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

P. K. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–158 (1996).
[CrossRef]

1994 (1)

B. A. Malomed and N. F. Smyth, “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994).
[CrossRef]

1993 (1)

D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[CrossRef]

1991 (1)

1990 (2)

Yu. S. Kivshar and V. V. Konotop, “Vector solitons in optical fiber with random birefringence,” Kvant. Elektron. (Moscow) 17, 1599–1602 (1990).

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fiber,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Abdullaev, F. Kh.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
[CrossRef]

Abdumalikov, A. A.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
[CrossRef]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

Andrekson, P.

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

Baizakov, B. B.

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
[CrossRef]

Bennion, I.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Bertilsson, K.

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

Blow, K. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Carter, G. M.

Doran, N. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Evangelides Jr., S. G.

Fedoruk, M. P.

Gabitov, I.

I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Georges, T.

Golovchenko, E. A.

Gordon, J. P.

Gornakova, A.

Hasegawa, A.

Holmes, P.

Jacob, J. M.

Karlsson, M.

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

Kath, W. L.

Kaup, D. J.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

T. Lakoba and D. J. Kaup, “Perturbation theory for Manakov system and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and V. V. Konotop, “Vector solitons in optical fiber with random birefringence,” Kvant. Elektron. (Moscow) 17, 1599–1602 (1990).

Knox, F. M.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Kodama, Y.

T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

Konotop, V. V.

Yu. S. Kivshar and V. V. Konotop, “Vector solitons in optical fiber with random birefringence,” Kvant. Elektron. (Moscow) 17, 1599–1602 (1990).

Kutz, J. N.

Lakoba, T.

T. Lakoba and D. J. Kaup, “Perturbation theory for Manakov system and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Lakoba, T. I.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Liang, A. H.

Malomed, B. A.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

B. A. Malomed and N. F. Smyth, “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994).
[CrossRef]

D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[CrossRef]

Maruta, A.

T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

Menyuk, C. R.

Mollenauer, L. F.

Okamavari, T.

T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

Pilipetskii, A. N.

Pincombe, A. H.

N. F. Smyth and A. H. Pincombe, “Effect of radiative loss on pulses in periodically inhomogeneous birefringent optical fibers,” Phys. Rev. E 57, 7231–7238 (1998).
[CrossRef]

Shapiro, E.

I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Smith, K.

Smith, N. J.

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

Smyth, N. F.

N. F. Smyth and A. H. Pincombe, “Effect of radiative loss on pulses in periodically inhomogeneous birefringent optical fibers,” Phys. Rev. E 57, 7231–7238 (1998).
[CrossRef]

B. A. Malomed and N. F. Smyth, “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994).
[CrossRef]

Tasgal, R. J.

D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[CrossRef]

Toda, H.

Turitsyn, S. K.

Turytsin, S.

S. Turytsin, “Stability of an optical soliton with Gaussian tails,” Phys. Rev. E 56, R3784–R3787 (1997).
[CrossRef]

I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

Ueda, T.

Wai, P. K.

P. K. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–158 (1996).
[CrossRef]

Yang, J.

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Zhang, X.

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

Electron. Lett. (2)

N. J. Smith, F. M. Knox, N. J. Doran, K. J. Blow, and I. Bennion, “Enhanced power solitons in optical fibers with periodic dispersion-management,” Electron. Lett. 32, 54–55 (1996).
[CrossRef]

X. Zhang, M. Karlsson, P. Andrekson, and K. Bertilsson, “Impact of polarization mode dispersion on dispersion-managed soliton systems,” Electron. Lett. 34, 1122–1124 (1998).
[CrossRef]

J. Lightwave Technol. (1)

P. K. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–158 (1996).
[CrossRef]

J. Opt. Soc. Am. B (4)

Kvant. Elektron. (Moscow) (1)

Yu. S. Kivshar and V. V. Konotop, “Vector solitons in optical fiber with random birefringence,” Kvant. Elektron. (Moscow) 17, 1599–1602 (1990).

Opt. Commun. (3)

F. Kh. Abdullaev, A. A. Abdumalikov, and B. B. Baizakov, “Propagation of chirped optical solitons in fibers with randomly varying parameters,” Opt. Commun. 138, 49–54 (1997).
[CrossRef]

T. Okamavari, A. Maruta, and Y. Kodama, “Reduction of Gordon–Haus jitter in a dispersion compensated optical transmission system: analysis,” Opt. Commun. 149, 261–266 (1998).
[CrossRef]

T. I. Lakoba, J. Yang, D. J. Kaup, and B. A. Malomed, “Conditions for stationary pulse propagation in the strong dispersion management regime,” Opt. Commun. 149, 366–375 (1998).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[CrossRef]

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fiber,” Phys. Rev. A 42, 563–571 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (6)

D. J. Kaup, B. A. Malomed, and R. J. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[CrossRef]

N. F. Smyth and A. H. Pincombe, “Effect of radiative loss on pulses in periodically inhomogeneous birefringent optical fibers,” Phys. Rev. E 57, 7231–7238 (1998).
[CrossRef]

B. A. Malomed and N. F. Smyth, “Resonant splitting of a vector soliton in a periodically inhomogeneous birefringent optical fiber,” Phys. Rev. E 50, 1535–1542 (1994).
[CrossRef]

I. Gabitov, E. Shapiro, and S. Turytsin, “Asymptotic breathing pulse in optical transmission system with dispersion compensation,” Phys. Rev. E 55, 3624–3633 (1997).
[CrossRef]

S. Turytsin, “Stability of an optical soliton with Gaussian tails,” Phys. Rev. E 56, R3784–R3787 (1997).
[CrossRef]

T. Lakoba and D. J. Kaup, “Perturbation theory for Manakov system and its application to pulse propagation in randomly birefringent fibers,” Phys. Rev. E 56, 6147–6165 (1997).
[CrossRef]

Other (4)

F. Kh. Abdullaev, J. Bronski, and G. Papanicolaou, “Soliton perturbations and random Kepler problem,” submitted to Physica D.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

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

J. Gukkenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

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Figures (6)

Fig. 1
Fig. 1

(a) Normalized intensity max|u(x, t)|, (b) chirp bu, and (c) square of relative distance between solitons r as a function of distance x for Δ0=0.2 and ω=2. Solid curves, ODE model; dashed curves, PDE model.

Fig. 2
Fig. 2

Phase-plane dynamics of (a) the PDE model and (b) the ODE model for Δ0=0.2 and ω=2.

Fig. 3
Fig. 3

Normalized mean values of the (a) intensity max|u(x, t)| and (b) chirp bu as a function of distance x for σ=0.3. Solid curves, ODE model; dashed curves: PDE model.

Fig. 4
Fig. 4

Mean square of the relative distance between solitons r2 as a function of distance x for (a) σ=0.2, (b) σ=0.3, (c) σ=0.4, and (d) σ=0.5. Solid curves, ODE model; dotted curves, PDE model; dashed lines, analytical estimation [formula (27)].

Fig. 5
Fig. 5

Mean length of vector soliton decay Lth as a function of random birefringence intensity σ.

Fig. 6
Fig. 6

Mean length of vector soliton decay Lth as a function of initial pulse energy E.

Equations (31)

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

 iux+iΔ(x)ut+D(x)2utt+(|u|2+A|v|2)u=0,
ivx-iΔ(x)vt+D(x)2vtt+(|v|2+A|u|2)v=0.
D(x)=d1+α1 m=1M δD(x-mxd),
Δ=πcΔnX0d¯λ,
Δ(x)=0,Δ(x1)Δ(x2)=B(x1-x2; lc).
Δ(x)=Δ0 sin(ωx).
L=i2(uxu*-ux*u)+iΔ(x)2(utu*-ut*u)+12|u|4-D(x)2|ut|2+i2(vxv*-vx*v)-iΔ(x)2(vtv*-vt*v)+12|v|4-D(x)2|vt|2+A|u|2|v|2,
u(x, t)=B1 exp-(1+ib1)(t-α1)22W12-iω1(t-α1)+iϕi,
v(x, t)=B2 exp-(1+ib2)(t-α2)22W22-iω2(t-α2)+iϕ2.
L¯=- L(x, t)dt.
L¯=i=12-Ei-bix4+biWix2Wi+ωiαix+ϕix+(-1)i-1Δ(x)Eiωi-D(x)2Ei1+bi22Wi2+ω2+Ei222πWi+AE1E2πWexp-r2W2.
δ0l L¯dx=0,
α1x=-D(x)ω1+Δ(x),α2x=-D(x)ω2-Δ(x),
E1ω1x=-E2ω2x=2AE1E2rπW3exp-r2W2,
WiWix=-D(x)bi,
bix=-D(x)(1+bi2)Wi2+Ei2πWi+2AEi+1Wi2πW31-2r2W2exp-r2W2,
rx=2Δ(x)-D(x)Ω,
Ωx=2AEπW3r exp-r2W2,
(W2)x=-2D(x)b,
rx=2Δ(x)-d¯Ω,
Ωx=2AEπWs3r exp-r2Ws2.
Ωxx+ω02Ω=aΔ(x),
ω02=2AEd¯πWs3,a=4AEπWs3.
Ω2=a2σ22ω02x-sin(2ω0x)2ω0.
DΩ=a2σ22ω02=8Aσ2πEWs3d¯,
r2πWs64A2E2Ωx2.
r24σ2x1-sin 2Ω0x2Ω0x.
Dr4σ2.
r2Δ0lc1/2F1/2,F=π3/4Ws9/22γ1/2A3/2E3/2d¯1/2.
Eπ1/2Ws322/3γ1/3Ad¯1/3.
u(x, t)=v(x, t)=1.363 exp(-0.296t2).

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