Abstract

Modulational instability in fiber Bragg gratings is investigated by coupled-mode theory. In the presence of anomalous dispersion the standard nonlinear Schrödinger results are obtained at low powers, although at higher powers, deviations from this well-known behavior occur. For normal dispersion an instability with a finite threshold is found. This instability has no equivalent in the nonlinear Schrödinger equation.

© 1998 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
  2. A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).
  3. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
    [CrossRef]
  4. J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [CrossRef] [PubMed]
  5. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
    [CrossRef] [PubMed]
  6. G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
    [CrossRef] [PubMed]
  7. A. S. Gouveia-Neto, M. E. Faldon, A. S. B. Bombra, P. G. J. Wigley, and J. R. Taylor, “Subpicosecond-pulse generation through cross-phase-modulation-induced modulational instability in optical fibers,” Opt. Lett. 13, 901–903 (1988).
    [CrossRef] [PubMed]
  8. B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
    [CrossRef]
  9. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  10. B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]
  11. C. Martijn de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
    [CrossRef]
  12. C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
    [CrossRef]
  13. H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
    [CrossRef]
  14. C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
    [CrossRef] [PubMed]
  15. A. B. Aceves, C. de Angelis, and S. Wabnitz, “Generation of solitons in a nonlinear periodic medium,” Opt. Lett. 17, 1566–1568 (1992).
    [CrossRef] [PubMed]
  16. R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
    [CrossRef]
  17. J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–598 (1993).
    [CrossRef]

1998 (1)

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

1997 (1)

1996 (2)

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

1995 (1)

1994 (1)

C. Martijn de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

1993 (1)

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–598 (1993).
[CrossRef]

1992 (2)

1991 (1)

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

1990 (3)

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Aceves, A. B.

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

A. B. Aceves, C. de Angelis, and S. Wabnitz, “Generation of solitons in a nonlinear periodic medium,” Opt. Lett. 17, 1566–1568 (1992).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

Armes, D. J.

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Bombra, A. S. B.

de Angelis, C.

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

de Sterke, C. Martijn

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

C. Martijn de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Faldon, M. E.

Feldman, S.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Feng, J.

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–598 (1993).
[CrossRef]

Froehlich, H.-G.

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

Gouveia-Neto, A. S.

Harvey, J. D.

S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
[CrossRef] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Kashyap, R.

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Kneubühl, F. K.

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–598 (1993).
[CrossRef]

Leonhardt, R.

S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
[CrossRef] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Murdoch, S. G.

Rothenberg, J.

J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

Sipe, J. E.

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

C. Martijn de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

Strasser, T. A.

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Swanton, A.

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

Taylor, J. R.

Wabnitz, S.

Wigley, P. G. J.

Winful, H. G.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Zakharov, V. E.

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Zamir, R.

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

H. G. Winful, R. Zamir, and S. Feldman, “Modulational instability in nonlinear periodic structures: implications for gap solitons,” Appl. Phys. Lett. 58, 1001–1003 (1991).
[CrossRef]

Electron. Lett. (2)

R. Kashyap, H.-G. Froehlich, A. Swanton, and D. J. Armes, “1.3-m Long super-step-chirped fibre Bragg grating with a continuous delay of 13.5 ns and bandwidth 10 nm for broadband dispersion compensation,” Electron. Lett. 32, 1807–1809 (1996).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, and J. E. Sipe, “Distributed feedback pulse generator based on nonlinear fibre grating,” Electron. Lett. 32, 2341–2342 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Feng and F. K. Kneubühl, “Solitons in a periodic structure with Kerr nonlinearity,” IEEE J. Quantum Electron. 29, 590–598 (1993).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

B. J. Eggleton, C. Martijn de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulation instability in high-birefringence fibers,” Opt. Commun. 78, 137–141 (1990).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (3)

C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A 42, 550–555 (1990).
[CrossRef]

C. M. de Sterke, “Stability analysis of nonlinear periodic media,” Phys. Rev. A 45, 8252–8258 (1992).
[CrossRef] [PubMed]

J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. P. Agrawal, “Modulational instability induced by cross-phase modulation,” Phys. Rev. Lett. 59, 880–883 (1987).
[CrossRef] [PubMed]

Prog. Opt. (1)

C. Martijn de Sterke and J. E. Sipe, “Gap solitons,” Prog. Opt. 33, 203–260 (1994).
[CrossRef]

Sov. Phys. JETP (1)

A. L. Berkhoer and V. E. Zakharov, “Self-excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486–490 (1970).

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).

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Figures (4)

Fig. 1
Fig. 1

Illustration of the significance of parameter f in the low-power limit.

Fig. 2
Fig. 2

Magnitude of the largest unstable wave number versus power for f=-0.5. Compared are low-intensity result q0 [Eq. (21)] (dotted curve), the first-order corrected version q0 from Eq. (A5) (long-dashed curve), and the exact numerical result (solid curve). For κ=500 m-1, λ=1.053×10-6 m, and a nonlinearity of 3.2×10-20 m2/W, a value P=0.1κ corresponds to P=26 GW/cm2.

Fig. 3
Fig. 3

Gain spectra showing the imaginary part of ω, ωi versus q, for two different cases: (a) f=-0.5, P=0.1κ; (b) f=-0.5, P=5κ. In both, the solid curves are numerical solutions of polynomial (10), and the dashed curves follow from nonlinear Schrödinger result (35).

Fig. 4
Fig. 4

Gain spectra showing ωi versus q, for two different cases: (a) f=0.5, P=0.5κ, and the most unstable mode is at infinity; (b) f=0.10, P=0.5κ, and the most unstable mode is finite, although the modes at infinity are unstable as well. The solid curves are exact solutions of polynomial (10), and the dashed curves follow from approximation (46).

Equations (69)

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i E±t±i E±z+κE+Γ(|E±|2+2|E|2)E±=0.
Γ=4πnλZ n(2).
E+=af2+1 exp[i(Qz-Ωt)]a+ exp[i(Qz-Ωt)],
E-=aff2+1 exp[i(Qz-Ωt)]a- exp[i(Qz-Ωt)],
Ω=-κ2 (f-1+f )-32Γa2,
Q=-κ2 (f-1-f )-12Γa2 1-f21+f2.
v=1-f21+f2.
E±=[a±+±(z, t)]exp[i(Qz-Ωt)],
i +t+i +z+κ--κf++
G[(+++)+2 f(-+-)]=0,
i -t-i -z+κ+-κf-1-+
G[2 f(+++)+f2(-+-)]=0,
G=Γa21+f2
±(z, t)=b±R cos(qz-ωt)+ib±I sin(qz-ωt),
(ω2-q2)2-2κ2(ω2-q2)-κ2f2(ω+q)2
-κ2f-2(ω-q)2+4κGf(3q2-ω2)=0,
P=2nZ (|E+|2+|E-|2)=2na2Z,
P=Γa2=Z2nΓ,P=G(1+f2),
P=2πλ n(2)P;
i ut+ω2 2uζ2+A|u|2u=0,
ζ=z-vt,
A=3-v22Γ,ω=(1-v2)3/2κ.
A=f-2+f2+4(f-1+f )2Γ,ω=8κ 1(f-1+f )3.
ω=ωq2 q2-(q0NLS)2,
q0NLS=4Au2|ω|.
0<|q| <q0NLS.
q0NLS=-κ2 (f-1+f )2(f-2+f2+4)fG.
ω2=q2+2κ(κ-G)±2κ2(κ-G)2+κq2(κ+2G).
q=0orq=±q0±12κG.
q=±qmax±3κG 2κ+Gκ+2G.
ω=±i q2κ 12κG-q2,
ω=2iκ(G-κ),
ω2=q2+2κ(κ+G)±2κ2(κ+G)2+κq2(κ-2G).
G>κ/2,
|q| >(κ+G)κ2G-κ,
ω=±q±iκ(2G-κ),
[ω-ω1(q)][ω-ω2(q)][ω-ω3(q)][ω-ω4(q)]
+4κGf(3q2-ω2)=0,
ω1,2=+κ2 (f-1+f )±q2-qκ(f-1-f )+κ24 (f-1+f )2,
ω3,4=-κ2 (f-1+f )±q2+qκ(f-1-f )+κ24 (f-1+f )2.
ω1,4=q f-1-ff-1+f4q2κ 1(f-1+f )3,
ω2,3=±κ(f-1+f ),
ω2ω3(ω-ω1)(ω-ω4)+4κGf(3q2-ω2)=0,
ω=q f-1-ff-1+f±4qκ 1(f-1+f )2×q2(f-1+f )2+κ2 Gf(f-2+f2+4),
ω=0orω=±κ2(f-1+f )2+4κGf.
G>-κ4 (f-1+f )2f.
y=1/q,δ=1/ω.
(y2-δ2)2-κ2y2δ2[2(y2-δ2)+f-2(y-δ)2+
f2(y+δ)2]+4κGfδ2y2(3δ2-y2)=0.
δ=±y+αy2+βy3+γy4.
α=κ2f2-2κGf,
Gf>κf2/2.
Gf>κ/(2 f2).
β=-α22-κ22 (1-3f2)-5κGf,
ωr=q+1q κ22+2κGf,
ωi=α1-κ2C4q2,
C=κ2(f2-1)+κ(f-1-f )(f-1-5f )Gf-18(Gf )22κGf-κ2f2.
G=κf2+ΔG.
ωi=α1-κ316q2 (2 f2+1)2ΔGf,
qminκ24 (2 f2+1) 1κΔGf.
q=q0NLS,ω=ω0f-1-ff-1+f q0NLS,
ω=ω0+ω,q=q0NLS+q,
uωω+uqq+uωωω2+uωqωq+uqqq2=0,
uω=+8κω0Gf(f-2+f2+3),
uq=-8κq0NLSGf(f-2+f2+1),
uωω=-κ2(f-1+f )2,
uωq=-2κ2(f2-f-2),
uqq=-κ2(f-1-f )2,
q0=q0NLS1+Gκ (f-1-f )2 (f-1+f)2 (f-2+f2+3)2f-2+f2+4,

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