Abstract

A perturbed nonlinear Schrödinger equation that describes femtosecond pulse propagation in spatially (axially) inhomogeneous optical fibers near the zero-dispersion point is considered. This equation, which has varying coefficients, is analyzed by means of a multiple-scale perturbation technique. Approximate analytical results, valid up to the first order, concerning both the envelope function and the carrier wave number and frequency, are derived. Necessary conditions for envelope bright solitary-wave formation, as well as the solutions themselves, are presented. Typical results concerning the effect of the inhomogeneity on the solitary-wave propagation also are given.

© 1995 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Orlando, Fla., 1989); A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, Berlin, 1989).
    [Crossref]
  2. P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, Opt. Lett. 11, 464 (1986); Opt. Lett. 12, 628 (1987); P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
    [Crossref] [PubMed]
  3. M. Desaix, D. Anderson, and M. Lisak, Opt. Lett. 15, 18 (1990); Opt. Lett. 15, 1285 (1990).
    [Crossref] [PubMed]
  4. G. P. Agrawal and M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
    [Crossref] [PubMed]
  5. Yu. S. Kivshar, Phys. Rev. A 43, 1677 (1991); Opt. Lett. 16, 892 (1991); Yu. S. Kivshar and V. V. Afanasjef, Phys. Rev. A 44, R1446 (1991).
    [Crossref] [PubMed]
  6. D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
    [Crossref]
  7. R. Grimshaw, Proc. R. Soc. London Ser. A 368, 377 (1979).
    [Crossref]
  8. H. H. Kuehl, J. Opt. Soc. Am. B 3, 709 (1988).
    [Crossref]
  9. Yu. S. Kivshar and V. V. Konotop, Sov. J. Quantum Electron. 19, 566 (1989).
    [Crossref]
  10. V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 73, 537 (1977) [Sov. Phys. JETP 46, 281 (1977)]; D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
    [Crossref]
  11. H. H. Chen and C. S. Liu, Phys. Fluids 21, 377 (1978).
    [Crossref]
  12. F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
    [Crossref]
  13. P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, IEEE J. Quantum Electron. 24, 373 (1988).
    [Crossref]
  14. Y. Kodama, J. Stat. Phys. 39, 597 (1985); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
    [Crossref]
  15. M. J. Potasek, J. Appl. Phys. 65, 941 (1989).
    [Crossref]
  16. M. J. Potasek, J. Appl. Phys. 63, 5186 (1988).
    [Crossref]
  17. N. Asano, Progr. Theor. Phys. (Japan) Suppl. 55, 52 (1974).
    [Crossref]
  18. N. J. Doran and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988); K. Kurokawa and M. Nakazawa, Appl. Phys. Lett. 58, 2871 (1991); M. Ding and K. Kikuchi, IEEE Photon. Technol. Lett. 4, 497 (1992).
    [Crossref]
  19. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge U. Press, Cambridge, 1990).

1995 (1)

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

1991 (1)

Yu. S. Kivshar, Phys. Rev. A 43, 1677 (1991); Opt. Lett. 16, 892 (1991); Yu. S. Kivshar and V. V. Afanasjef, Phys. Rev. A 44, R1446 (1991).
[Crossref] [PubMed]

1990 (1)

1989 (3)

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

M. J. Potasek, J. Appl. Phys. 65, 941 (1989).
[Crossref]

Yu. S. Kivshar and V. V. Konotop, Sov. J. Quantum Electron. 19, 566 (1989).
[Crossref]

1988 (4)

H. H. Kuehl, J. Opt. Soc. Am. B 3, 709 (1988).
[Crossref]

M. J. Potasek, J. Appl. Phys. 63, 5186 (1988).
[Crossref]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, IEEE J. Quantum Electron. 24, 373 (1988).
[Crossref]

N. J. Doran and D. Wood, J. Opt. Soc. Am. B 5, 1301 (1988); K. Kurokawa and M. Nakazawa, Appl. Phys. Lett. 58, 2871 (1991); M. Ding and K. Kikuchi, IEEE Photon. Technol. Lett. 4, 497 (1992).
[Crossref]

1986 (2)

1985 (1)

Y. Kodama, J. Stat. Phys. 39, 597 (1985); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[Crossref]

1979 (1)

R. Grimshaw, Proc. R. Soc. London Ser. A 368, 377 (1979).
[Crossref]

1978 (1)

H. H. Chen and C. S. Liu, Phys. Fluids 21, 377 (1978).
[Crossref]

1977 (1)

V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 73, 537 (1977) [Sov. Phys. JETP 46, 281 (1977)]; D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[Crossref]

1974 (1)

N. Asano, Progr. Theor. Phys. (Japan) Suppl. 55, 52 (1974).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
[Crossref] [PubMed]

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

Anderson, D.

Asano, N.

N. Asano, Progr. Theor. Phys. (Japan) Suppl. 55, 52 (1974).
[Crossref]

Bass, F. G.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

Belia, I.

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

Chen, H. H.

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, IEEE J. Quantum Electron. 24, 373 (1988).
[Crossref]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, Opt. Lett. 11, 464 (1986); Opt. Lett. 12, 628 (1987); P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
[Crossref] [PubMed]

H. H. Chen and C. S. Liu, Phys. Fluids 21, 377 (1978).
[Crossref]

Desaix, M.

Doran, N. J.

Frantzeskakis, D. J.

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

Grimshaw, R.

R. Grimshaw, Proc. R. Soc. London Ser. A 368, 377 (1979).
[Crossref]

Hizanidis, K.

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

Infeld, E.

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge U. Press, Cambridge, 1990).

Karpman, V. I.

V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 73, 537 (1977) [Sov. Phys. JETP 46, 281 (1977)]; D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[Crossref]

Kivshar, Yu. S.

Yu. S. Kivshar, Phys. Rev. A 43, 1677 (1991); Opt. Lett. 16, 892 (1991); Yu. S. Kivshar and V. V. Afanasjef, Phys. Rev. A 44, R1446 (1991).
[Crossref] [PubMed]

Yu. S. Kivshar and V. V. Konotop, Sov. J. Quantum Electron. 19, 566 (1989).
[Crossref]

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

Kodama, Y.

Y. Kodama, J. Stat. Phys. 39, 597 (1985); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[Crossref]

Konotop, V. V.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

Yu. S. Kivshar and V. V. Konotop, Sov. J. Quantum Electron. 19, 566 (1989).
[Crossref]

Kuehl, H. H.

H. H. Kuehl, J. Opt. Soc. Am. B 3, 709 (1988).
[Crossref]

Lee, Y. C.

Lisak, M.

Liu, C. S.

H. H. Chen and C. S. Liu, Phys. Fluids 21, 377 (1978).
[Crossref]

Maslov, E. M.

V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 73, 537 (1977) [Sov. Phys. JETP 46, 281 (1977)]; D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[Crossref]

Menyuk, C. R.

Potasek, M. J.

M. J. Potasek, J. Appl. Phys. 65, 941 (1989).
[Crossref]

M. J. Potasek, J. Appl. Phys. 63, 5186 (1988).
[Crossref]

G. P. Agrawal and M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
[Crossref] [PubMed]

Pritula, G. M.

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

Rowlands, G.

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge U. Press, Cambridge, 1990).

Tombras, G. S.

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

Wai, P. K. A.

Wood, D.

IEEE J. Quantum Electron. (2)

D. J. Frantzeskakis, K. Hizanidis, G. S. Tombras, and I. Belia, IEEE J. Quantum Electron. 31, 183 (1995).
[Crossref]

P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, IEEE J. Quantum Electron. 24, 373 (1988).
[Crossref]

J. Appl. Phys. (2)

M. J. Potasek, J. Appl. Phys. 65, 941 (1989).
[Crossref]

M. J. Potasek, J. Appl. Phys. 63, 5186 (1988).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Stat. Phys. (1)

Y. Kodama, J. Stat. Phys. 39, 597 (1985); Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. QE-23, 510 (1987).
[Crossref]

Opt. Commun. (1)

F. G. Bass, Yu. S. Kivshar, V. V. Konotop, and G. M. Pritula, Opt. Commun. 70, 309 (1989); R. Scharf and A. R. Bishop, Phys. Rev. A 46, R2973 (1992).
[Crossref]

Opt. Lett. (2)

Phys. Fluids (1)

H. H. Chen and C. S. Liu, Phys. Fluids 21, 377 (1978).
[Crossref]

Phys. Rev. A (2)

G. P. Agrawal and M. J. Potasek, Phys. Rev. A 33, 1765 (1986).
[Crossref] [PubMed]

Yu. S. Kivshar, Phys. Rev. A 43, 1677 (1991); Opt. Lett. 16, 892 (1991); Yu. S. Kivshar and V. V. Afanasjef, Phys. Rev. A 44, R1446 (1991).
[Crossref] [PubMed]

Proc. R. Soc. London Ser. A (1)

R. Grimshaw, Proc. R. Soc. London Ser. A 368, 377 (1979).
[Crossref]

Progr. Theor. Phys. (Japan) Suppl. (1)

N. Asano, Progr. Theor. Phys. (Japan) Suppl. 55, 52 (1974).
[Crossref]

Sov. J. Quantum Electron. (1)

Yu. S. Kivshar and V. V. Konotop, Sov. J. Quantum Electron. 19, 566 (1989).
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

V. I. Karpman and E. M. Maslov, Zh. Eksp. Teor. Fiz. 73, 537 (1977) [Sov. Phys. JETP 46, 281 (1977)]; D. J. Kaup and A. C. Newell, Proc. R. Soc. London Ser. A 361, 413 (1978).
[Crossref]

Other (2)

E. Infeld and G. Rowlands, Nonlinear Waves, Solitons, and Chaos (Cambridge U. Press, Cambridge, 1990).

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

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Figures (5)

Fig. 1
Fig. 1

Normalized (to the maximum amplitude value in the homogeneous case, r0) envelope solitary solution as a function of time (in femtoseconds) for C1 > 0 and several values of ∊C2: the homogeneous case (∊C2 = 0), the extreme case of overall flipping of the mode (−2), the symmetrically modified mode (−1), two symmetric sign-preserving limiting cases ( C 2 = y ± = ± 3 / 2 - 1), and an indicative weakly inhomogeneous case [−0.3 (C2 = −3, = 0.1)]. The last-named value is shown as a dashed curve.

Fig. 2
Fig. 2

Displacement of (in femtoseconds) of the positive and negative peaks for the full ∊C2 interval for C1 > 0.

Fig. 3
Fig. 3

Normalized (to the maximum amplitude value in the homogeneous case, r0) envelope solitary solution as a function of time (in femtoseconds) for C1 < 0 and several values of ∊C2: the homogeneous case (∊C2 = 0), the extreme case of overall flipping of the mode (−2), the symmetrically modified mode (−1), two symmetric sign-preserving limiting cases ( C 2 = y ± = ± 3 / 2 - 1), and an indicative weakly inhomogeneous case [−0.3 (C2 = −3, = 0.1)]. The last-named value is shown as a dashed curve.

Fig. 4
Fig. 4

Displacement (in femtoseconds) of the positive and negative peaks for the full ∊C2 interval for C1 < 0.

Fig. 5
Fig. 5

Normalized peak values for the full ∊C2 interval irrespective of the sign of the parameter C1.

Equations (56)

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i u z - k 2 2 u t r 2 + b 1 u u 2 - i k 6 3 u t r 3 + i b 2 t r ( u u 2 ) + i b 3 u t r ( u 2 ) = 0 ,
i u z - 1 2 ( k + Δ k ) 2 u t r 2 + ( b 1 + Δ b 1 ) u u 2 - i 1 6 ( k + Δ k ) 3 u t r 3 + i ( b 2 + Δ b 2 ) t r ( u u 2 ) + i ( b 3 + Δ b 3 ) u t r ( u 2 ) = 0 ,
q = t 0 3 / 2 ( 6 b 1 k ) 1 / 2 u ,             ζ = k 6 t 0 3 z ,             τ = t r / t 0
i q ζ - f 1 2 q τ 2 + f 2 q q 2 - i f 3 3 q τ 3 + i f 4 τ ( q q 2 ) + i f 5 q τ ( q 2 ) = 0.
f 1 = 3 t 0 Δ k k ,             f 2 = 1 + Δ b 1 b 1 , f 3 = - 1 + Δ k k ,             f 4 = χ + Δ χ ,             f 5 = δ + Δ δ ,
Z = ζ ,             T = τ ,
f 1 = F 1 ( Z ) ,             f 2 = 1 + F 2 ( Z ) ,             f 3 = - 1 + F 3 ( Z ) , f 4 = χ + F 4 ( Z ) ,             f 5 = δ + F 5 ( Z ) ,
q = r e i ϕ = ( r 0 + r 1 + 2 r 2 + ) e i ϕ ,
θ = - 1 Θ = - 1 Θ 0 + Θ 1 + Θ 2 + ,
ϕ = - 1 Φ = - 1 Φ 0 + Φ 1 + Φ 2 + .
θ τ = - t 0 Ω = - t 0 ( Ω 0 + Ω 1 + 2 Ω 2 + ) ,
θ ζ = L T ( K - k Ω ) = t 0 [ ( K 0 - k 0 Ω 0 ) + ( K 1 - k 1 Ω 0 - k 0 Ω 1 ) + ] ,
ϕ τ = - t 0 ω ^ = - t 0 ( ω ^ 0 + ω ^ 1 + 2 ω ^ 2 + ) ,
ϕ ζ = L T ( k ^ - k ω ^ ) = L T [ ( k ^ 0 - k 0 ω ^ 0 ) + ( k ^ 1 - k 1 ω ^ 0 - k 0 ω ^ 1 ) + ] ,
3 t 0 3 Ω 0 2 ω ^ 0 2 r 0 θ 2 - [ L T ( k ^ 0 - k 0 ω ^ 0 ) + t 0 3 ω ^ 0 3 ] r 0 + ( 1 + χ t 0 ω ^ 0 ) r 0 3 = 0 ,
t 0 3 Ω 0 3 3 r 0 θ 3 - [ L T ( K 0 - k 0 Ω 0 ) + 3 t 0 3 Ω 0 ω ^ 0 2 ] r 0 θ + ( 3 χ + 2 δ ) t 0 Ω 0 r 0 2 r 0 θ = 0.
Ω 0 3 ω ^ 0 = L T ( K 0 - k 0 Ω 0 ) + 3 t 0 3 Ω 0 ω ^ 0 2 L T ( k ^ 0 - k 0 ω ^ 0 ) + t 0 3 ω ^ 0 3 = ( 3 χ + 2 δ ) t 0 Ω 0 3 ( 1 + χ t 0 ω ^ 0 ) .
ω ^ 0 = 1 2 t 0 ( χ + δ ) ,
k ^ 0 = ( 3 K 0 Ω 0 - 2 k 0 ) ω ^ 0 + 8 t 0 3 L T ω ^ 0 3 .
K 0 Ω 0 = - ( q / z ) 0 ( q / t ) 0 .
( 1 V c ) 0 = L T t 0 ( k 0 - k ^ 0 ω ^ 0 ) ,             ( 1 V e ) 0 = L T t 0 ( k 0 - K 0 Ω 0 ) .
2 r 0 θ 2 + α r 0 + β r 0 3 = 0 ,
α = - 1 3 Ω 0 2 [ ω ^ 0 2 - 1 t 0 2 ( 1 V c ) 0 ] ,
β = 3 χ + 2 δ 3 t 0 2 Ω 0 2 .
1 2 ( r 0 θ ) 2 + α 2 r 0 2 + β 4 r 0 4 = C ( Z , T ) ,
r 0 = ( 2 α β ) 1 / 2 sech [ α 1 / 2 ( θ - θ 0 ) ] ,
( q / z ) 0 ( q / t ) 0 < t 0 4 L T ( χ + δ ) 2 - k 0 .
Γ 1 2 r 1 θ 2 + Γ 2 r 1 + Γ 3 r 0 2 r 1 = Γ 4 2 r 0 θ 2 + Γ 5 r 0 + Γ 6 r 0 3 ,
θ ( γ 1 2 r 1 θ 2 + γ 2 r 1 + γ 3 r 0 2 r 1 ) = θ ( γ 4 2 r 0 θ 2 + γ 5 r 0 + γ 6 r 0 3 ) ,
Ω 1 = Ω 0 ( ω ^ 1 ω ^ 0 - F 1 3 t 0 ω ^ 0 ) .
ω ^ 1 = 1 2 ( χ + δ ) [ F 2 + 3 χ + 2 δ 3 t 0 F 1 - 2 ω ^ 0 ( F 4 + F 5 ) ] .
K 1 = Ω 0 3 ω ^ 0 k ^ 1 + [ k 0 Ω 1 + 2 3 k 1 Ω 0 - Ω 0 k 0 ω ^ 1 3 ω ^ 0 + 8 t 0 2 Ω 0 ω ^ 0 L T × ( 1 3 F 1 + 1 3 t 0 ω ^ 0 F 3 - t 0 ω ^ 1 ) ] .
1 α 2 r 1 θ 2 - { 1 - 6 sech 2 [ α 1 / 2 ( θ - θ 0 ) ] } r 1 = G ( r 0 ) : = E 1 r 0 - E 2 r 0 3 ,
R 1 = sech ( x ) tanh ( x ) ,
R 2 = x sech ( x ) tanh ( x ) + cosh ( x ) - sech ( x ) ,
r 1 = c 1 R 1 + c 2 R 2 - R 1 0 θ G ( r 0 ) R 2 W ( R 1 , R 2 ) d θ + R 2 0 θ G ( r 0 ) R 1 W ( R 1 , R 2 ) d θ ,
r 1 = { c 1 + [ c 2 + 1 4 α ( E 1 - 3 2 E 2 ) ] x } sech ( x ) tanh ( x ) + [ c 2 3 + 1 4 α ( E 1 + 1 2 E 2 ) tanh 2 ( x ) ] cosh ( x ) - c 2 sech ( x ) .
2 E 1 + 3 E 2 = 0 ,
c 2 = 3 E 2 4 α .
r 1 = c 1 sech ( x ) tanh ( x ) - E 2 α sech ( x ) .
r = ( 2 α β ) 1 / 2 ( sech [ α 1 / 2 ( θ - θ 0 ) ] + ( β 2 ) 1 / 2 1 α × { E 2 α 1 / 2 sech [ α 1 / 2 ( θ - θ 0 ) ] - 1 Ω 0 ( q t ) 0 × sech [ α 1 / 2 ( θ - θ 0 ) ] tanh [ α 1 / 2 ( θ - θ 0 ) ] } ) .
r ( θ ) r 0 = { 1 + C 2 - sign ( C 1 ) [ - 3 C 2 ( C 2 + 2 ) ] 1 / 2 tanh ( x ) } × sech ( x ) + O ( n ; n > 1 ) ,
Γ 1 = 3 t 0 3 Ω 0 2 ω ^ 0 ,
Γ 2 = - L T ( k ^ 0 - k 0 ω ^ 0 ) - t 0 3 ω ^ 0 3 ,
Γ 3 = 3 ( 1 + χ t 0 ω ^ 0 ) ,
Γ 4 = t 0 Ω 0 [ t 0 Ω 0 ( F 1 + 3 t 0 ω ^ 0 F 3 ) - 3 t 0 2 ( Ω 0 ω ^ 1 + 2 Ω 1 ω ^ 0 ) ]
Γ 5 = - t 0 2 ω ^ 0 2 ( F 1 + t 0 ω ^ 0 F 3 ) + L T ( k ^ 1 - k 0 ω ^ 1 - k 1 ω ^ 0 ) + 3 t 0 3 ω ^ 0 2 ω ^ 1 ,
Γ 6 = - F 2 - χ t 0 ω ^ 1 - t 0 ω ^ 0 F 4 ,
γ 1 = t 0 3 Ω 0 3 ,
γ 2 = - L T ( K 0 - k 0 Ω 0 ) - 3 t 0 3 Ω 0 ω ^ 0 2 ,
γ 3 = ( 3 χ + 2 δ ) t 0 Ω 0
γ 4 = t 0 3 Ω 0 2 ( Ω 0 F 3 - 3 Ω 1 ) ,
γ 5 = L T ( K 1 - k 0 Ω 1 - k 1 Ω 0 ) - 2 t 0 2 Ω 0 ω ^ 0 F 1 - 3 t 0 3 ω ^ 0 [ ω ^ 0 ( Ω 0 F 3 - Ω 1 ) - 2 Ω 0 ω ^ 1 ] ,
γ 6 = - t 0 [ 3 ( χ Ω 1 + Ω 0 F 4 ) + 2 ( δ Ω 1 + Ω 0 F 5 ) ] ,
E 1 = 1 3 t 0 Ω 0 ω ^ 0 [ Ω 0 F 1 + 3 t 0 Ω 0 ω ^ 0 F 0 - 3 t 0 Ω 0 ω ^ 1 - 6 t 0 Ω 1 ω ^ 0 ] - 1 3 t 0 Ω 0 2 ω ^ 0 α × [ ω ^ 0 2 F 1 + t 0 ω ^ 0 3 F 3 - 3 t 0 ω ^ 0 2 ω ^ 1 - L T t 0 2 ( k ^ 1 - k 0 ω ^ 1 - k 1 ω ^ 0 ) ] ,
E 2 = b 3 t 0 Ω 0 ω ^ 0 α ( Ω 0 F 1 + 3 t 0 Ω 0 ω ^ 0 F 3 - 3 t 0 Ω 0 ω ^ 1 - 6 t 0 Ω 1 ω ^ 0 ) + 1 3 t 0 3 Ω 0 2 ω ^ 0 α × ( F 2 + χ t 0 ω ^ 1 + t 0 ω ^ 0 F 4 ) .

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