Abstract

The stability of local structures in optical systems is of great importance. We demonstrate that using the Floquet–Fourier–Hill (FFH) method provides a substantial improvement in both speed and accuracy over finite-difference methods that are commonly used. Furthermore, we show how to incorporate the effect of nonlocal saturable gain in the linearization and stability predictions. Several examples of problems are worked with both the FFH and finite-difference methods and compared in the context of mode-locked laser models.

© 2010 Optical Society of America

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References

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  1. P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, 1981).
  2. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Springer-Verlag, 2003).
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    [CrossRef]
  7. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
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  8. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099–2111 (1997).
    [CrossRef]
  9. T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master-mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
    [CrossRef]
  10. T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
    [CrossRef]
  11. B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
    [CrossRef]
  12. B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
    [CrossRef]
  13. G. W. Hill, “On the part of the lunar perigee which is a function of the mean motions of the sun and the moon,” Acta Math. 8, 1–36 (1886).
    [CrossRef]
  14. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  15. B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, 1998).
  16. L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000).
    [CrossRef]
  17. G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Ann. Sci. Ec. Normale Super. 2, 47–89 (1883).
  18. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).
  19. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).
  20. E. Farnum and J. N. Kutz, “Multi-frequency mode-locked lasers,” J. Opt. Soc. Am. B 25, 1002–1010 (2008).
    [CrossRef]
  21. B. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
    [CrossRef]
  22. J. N. Kutz, “Passive mode-locking using phase-sensitive amplification,” Phys. Rev. A 78, 013845 (2008).
    [CrossRef]
  23. B. Bale, E. Farnum, and J. N. Kutz, “Dynamics of multifrequency mode-locking driven by homogenous and inhomogenous gain broadening effects,” J. Opt. Soc. Am. B 25, 1479–1487 (2008).
    [CrossRef]
  24. N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).
  25. J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
    [CrossRef] [PubMed]
  26. J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
    [CrossRef] [PubMed]
  27. B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
    [CrossRef]

2009 (1)

B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
[CrossRef]

2008 (5)

2007 (1)

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

2006 (2)

B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[CrossRef]

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

2005 (1)

2004 (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[CrossRef]

2002 (1)

2000 (1)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

1997 (1)

1991 (1)

1886 (1)

G. W. Hill, “On the part of the lunar perigee which is a function of the mean motions of the sun and the moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

1883 (1)

G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Ann. Sci. Ec. Normale Super. 2, 47–89 (1883).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).

Bale, B.

B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
[CrossRef]

B. Bale, E. Farnum, and J. N. Kutz, “Dynamics of multifrequency mode-locking driven by homogenous and inhomogenous gain broadening effects,” J. Opt. Soc. Am. B 25, 1479–1487 (2008).
[CrossRef]

B. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

Carter, J.

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

Chen, F.

F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, 1984).

Coddington, E. A.

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Deconinck, B.

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[CrossRef]

Drazin, P. G.

P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, 1981).

Farnum, E.

Floquet, G.

G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Ann. Sci. Ec. Normale Super. 2, 47–89 (1883).

Fornberg, B.

B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, 1998).

Fujimoto, J. G.

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, 1990).

Haus, H. A.

Hill, G. W.

G. W. Hill, “On the part of the lunar perigee which is a function of the mean motions of the sun and the moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Ippen, E. P.

Kapitula, T.

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master-mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[CrossRef]

Kiyak, F.

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

Kutz, J. N.

B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
[CrossRef]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
[CrossRef] [PubMed]

B. Bale, E. Farnum, and J. N. Kutz, “Dynamics of multifrequency mode-locking driven by homogenous and inhomogenous gain broadening effects,” J. Opt. Soc. Am. B 25, 1479–1487 (2008).
[CrossRef]

B. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

J. N. Kutz, “Passive mode-locking using phase-sensitive amplification,” Phys. Rev. A 78, 013845 (2008).
[CrossRef]

E. Farnum and J. N. Kutz, “Multi-frequency mode-locked lasers,” J. Opt. Soc. Am. B 25, 1002–1010 (2008).
[CrossRef]

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[CrossRef]

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013–2015 (2005).
[CrossRef] [PubMed]

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master-mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[CrossRef]

Levinson, N.

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).

Murray, J. D.

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Springer-Verlag, 2003).

Namiki, S.

Proctor, J.

Reid, W. H.

P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, 1981).

Sandstede, B.

B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
[CrossRef]

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636–650 (2008).
[CrossRef] [PubMed]

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master-mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
[CrossRef]

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000).
[CrossRef]

Yu, C. X.

Acta Math. (1)

G. W. Hill, “On the part of the lunar perigee which is a function of the mean motions of the sun and the moon,” Acta Math. 8, 1–36 (1886).
[CrossRef]

Ann. Sci. Ec. Normale Super. (1)

G. Floquet, “Sur les équations différentielles linéaires à coefficients périodiques,” Ann. Sci. Ec. Normale Super. 2, 47–89 (1883).

IEEE J. Quantum Electron. (1)

B. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multifrequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. 44, 976–983 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

B. Bale, J. N. Kutz, and B. Sandstede, “Optimizing waveguide array mode-locking for high-power fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 15, 220–231 (2009).
[CrossRef]

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

Indiana Univ. Math. J. (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095–1126 (2004).
[CrossRef]

J. Comput. Phys. (1)

B. Deconinck and J. N. Kutz, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[CrossRef]

J. Opt. Soc. Am. B (5)

Math. Comput. Simul. (1)

B. Deconinck, F. Kiyak, J. Carter, and J. N. Kutz, “SpectrUW: a laboratory for the numerical exploration of spectra of linear operators,” Math. Comput. Simul. 74, 370–378 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

J. N. Kutz, “Passive mode-locking using phase-sensitive amplification,” Phys. Rev. A 78, 013845 (2008).
[CrossRef]

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

Other (10)

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman and Hall, 1997).

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge Univ. Press, 1998).

L. N. Trefethen, Spectral Methods in Matlab (SIAM, 2000).
[CrossRef]

P. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge Univ. Press, 1981).

J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications (Springer-Verlag, 2003).

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, 1990).

F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, 1984).

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Figures (4)

Fig. 1
Fig. 1

Bifurcation diagram of the η   sech   η t solution branch calculated using the FFH algorithm. Spectrally stable segments are shown as solid curve and unstable segments are dotted curve. Representative spectra and supporting numerical simulations are shown for four values of g 0 . In each case the simulations agree with the conclusion drawn from the calculation of the spectrum.

Fig. 2
Fig. 2

The importance of including saturable gain is shown: in this case, the η   sech   ω t solution is always unstable for fixed gain. Furthermore it can be seen that the FFH and FD methods agree qualitatively, but the FFH method provides better resolution of the crucially important eigenvalues, namely, the zero eigenvalues and the eigenvalue with the largest real part. The dots (crosses) indicate eigenvalues calculated using FFH (FD) method. For each method the spectrum was calculated using a 255 × 255 matrix.

Fig. 3
Fig. 3

The spectral accuracy of the FFH method is demonstrated by considering the resolution of the two zero eigenvalues associated with translational symmetry and phase invariance plotted versus the matrix size n. Both zero eigenvalues have converged to numerical precision well before n = 500 . By comparison, the FD method (dotted-dashed curve) converges very slowly to the zero eigenvalues.

Fig. 4
Fig. 4

The WGA undergoes a Hopf bifurcation as g 0 increases past 2.4. Although the eigenvalues are seen to cross the real axis when calculated using both the FFH (dots) and FD (crosses) methods, only the FFH method accurately resolves the zero mode. In this case, the FD method would make an incorrect prediction about the stability since the “zero” eigenvalue registers as positive, when in fact this is just a numerical artifact. Simulations (top panels) lend further support to the stability properties indicated by spectra (middle panels) calculated by the FFH method.

Tables (4)

Tables Icon

Table 1 Comparison Between the FFH and FD Calculations for the Phase Sensitive Amplifier Linear Operator a, b

Tables Icon

Table 2 Comparison of Eigenvalue with the Maximum Real Part Using FFH and FD Methods for Varying Grid Sizes a, b

Tables Icon

Table 3 Comparison of the Two Zero Eigenvalues Using the FFH and FD Methods for Varying Grid Sizes a, b

Tables Icon

Table 4 Comparison of the Two Zero Eigenvalues using the FFH and FD Methods for Varying Grid Sizes of the Waveguide Array for Parameters Corresponding to η = 1.83 a, b, c

Equations (64)

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u z = F ( u ( t , z ) ) ,
u ( t , z ) = ( U ( t ) + v ( t , z ) ) ,
v z = L [ U ( t ) ] v ,
L [ U ( t ) ] w ( t ) = λ w ( t ) .
u z = k = 0 M f k ( u ( t , z ) ) t k u ( t , z ) + G ( u ) ( u ( t , z ) ) ,
G ( u ) = 2 g 0 1 + u 2 / e 0 ,
v z = k = 0 M f k ( U ) t k v + g sat ( v ) + g pert ( U ) [ U ( t ) v ( t ) + U ( t ) v ( t ) ] d t + O ( v 2 ) ,
g sat = 2 g 0 1 + U 2 ,
g pert = g sat 1 + U 2 .
v z = k = 0 M f k t k v + F ( t ) [ U ( t ) v ( t , z ) + U ( t ) v ( t , z ) ] d t ,
L [ U ] = L [ U ] + K [ U ] ,
L = k = 0 M f k ( t ) t k ,
K [ U ] w ( t ) = F ( t , U ( t ) ) U ( t ) w ( t ) d t .
w ( t ) = ϕ ( t ) e i μ t ,     μ [ 0 , 2 π / L ) ,
L [ U ] w ( t ) = λ w ( t ) ( L ̂ + K ̂ ) ϕ ̂ = λ ϕ ̂ ,
ϕ ( t ) = m = ϕ ̂ m e i k m t ,
ϕ ̂ m = 1 P L P L / 2 P L / 2 ϕ ( x ) e i k m t d t ,
λ ϕ ̂ n = λ 1 L ϕ ( x ) e i k n t d t = 1 L ( λ w ( t ) ) e i k n t i μ t d t = 1 L L [ U ] w ( t ) e i k n t i μ t d t = 1 L e i k n t i μ t ( L + K [ U ] ) ϕ ( t ) e i μ t d t = m = 1 L e i k n t i μ t ( L + K [ U ] ) ϕ m e i k m t + i μ t d t ,
( L ̂ ( μ ) ϕ ̂ ) n = m = k = 0 M f ̂ k , n m [ i ( μ + 2 π m / L ) ] k ϕ ̂ m ,
L ̂ n m = k = 0 M f ̂ k , n m [ i ( μ + 2 π m / L ) ] k .
m = K ̂ n m ( μ ) ϕ ̂ m = λ ϕ ̂ n m = L ̂ n m ϕ ̂ m = 1 L e i μ t i k n t ( F ( t ) U ( y ) w ( y ) d y ) d t = 1 L e i ( μ + k n ) t r F ̂ r e i k r t ( p U ̂ p e i k p y m ϕ ̂ m e i ( k m + μ ) y d y ) d t = 1 L r , p , m F ̂ r U ̂ p ϕ ̂ m e i t ( μ + k n k r ) ( e i y ( k m + k p + μ ) d y ) d t ,
m = K ̂ n m ( 0 ) ϕ ̂ m = 1 L r , s , m F ̂ r U ̂ s ϕ ̂ m e i t ( k n k r ) ( e i y ( k m + k s ) d y ) d t = 1 L r , s , m L δ 0 , k m k s F ̂ r U ̂ s ϕ ̂ m e i ( k m + k s k n + k r ) t d t = L r , s , m F ̂ r U ̂ s ϕ ̂ m δ 0 , ( m + s ) δ s , n m r ,
K ̂ ϕ ̂ n = m ( L F ̂ n U ̂ m ) ϕ ̂ m .
K ̂ n m = L F ̂ n U ̂ m .
L ̂ ϕ ̂ = λ ϕ ̂ ,
L ̂ n m = ( k = 0 M f ̂ k , n m [ i ( 2 π m / L ) ] k ) + K ̂ n m .
u z = ( 2 g 0 1 + u 2 g 0 ) u ,
v z = ( 2 g 0 1 + U 2 g 0 ) v 4 g 0 ( 1 + U 2 ) 2 U ( t ) U ( y ) v ( y , z ) d y .
L [ U ] = ( 2 g 0 1 + U 2 g 0 ) 4 g 0 ( 1 + U 2 ) 2 K [ U ] .
L ̂ n m = g 0 δ n , m ,
L ̂ n m = g 0 δ n , 0 δ m , 0 ,
F ( t ) a b U ( t ) v ( t ) d t ( F U T M ) v ,
M = h 3 [ 2 0 0 0 0 4 0 0 0 0 2 4 0 0 0 2 ] .
u z + 1 4 ( t 2 ω ) 2 u γ ( z ) u ω u 3 + u 5 + 3 δ β u ( u t ) 2 + δ ( 1 + β ) u 2 u t t = 0 ,
γ ( z ) = 2 g 0 1 + u 2 .
U ( t ) = η   sech ( η t ) ,
L [ U ] = 1 4 t 4 + ( ω 2 δ ( 1 + β ) U 2 ) t 2 6 δ β U U t t + ( g sat ω 2 4 + 3 ω U 2 5 U 4 3 β δ U t 2 2 δ ( 1 + β ) U U t t ) + K [ U ] ,
K [ U ] v = 2 g pert U ( t ) U ( t ) v ( t , z ) d t ,
L [ U ( t ) ] V ( t ) = L V ( t ) + F ( t ) U T ( t ) V ( t ) d t ,
V ( t ) = [ R ( t ) I ( t ) ] ,
U ( t ) = [ U R ( t ) U I ( t ) ] ,
F ( t ) = [ F R ( t ) F I ( t ) ] ,
L ̂ = L ̂ + K ̂ ,
L ̂ = [ L ̂ 11 L ̂ 12 L ̂ 21 L ̂ 22 ] ,
K ̂ = [ M 0 0 M ] F ̂ U ̂ T [ M 0 0 M ] ,
i ( u n z + δ n u n t ) + 1 2 2 u n t 2 + ( 1 i β n ) | u n | 2 u n i g n ( z ) ( 1 + τ 2 t 2 ) u n + 2 ( | u n 1 | 2 + | u n + 1 | 2 ) u n + i σ n | u n | 4 u n + i γ u n = 0 ,
u ( z , t ) = η   sech ( ω t ) 1 + i A e i Θ z = u 0 ( t ) e i Θ z .
u ( z , t ) = ( u 0 ( t ) + v ( z , t ) ) e i Θ z = ( ( U R ( t ) + i U I ( t ) ) + ( V R ( t ) + i V I ( t ) ) e λ z ) e i Θ z ,
L V = [ τ g sat 1 2 1 2 τ g sat ] t 2 V + [ F + ( U R , U I ) G ( U R , U I ) G + ( U I , U R ) F ( U I , U R ) ] V 2 g sat 1 + U 2 ( U T ( t ) V ( t ) d x ) ( 1 + τ 2 t 2 ) U ( t ) ,
F ± ( v , w ) = β ( 3 v 2 + w 2 ) 2 v w ( γ g sat ) ,
G ± ( v , w ) = ± ( 3 v 2 + w 2 ) + 2 β v w Θ ,
i A 0 z + 1 2 2 A 0 t 2 + β 0 | A 0 | 2 A 0 + C A 1 + i γ 0 A 0 i g ( z ) ( 1 + τ 2 t 2 ) A 0 = 0 ,
i A 1 z + C ( A 2 + A 0 ) + i γ 1 A 1 = 0 ,
i A 2 z + C A 1 + i γ 2 A 2 = 0 ,
g ( z ) = 2 g 0 1 + A 0 2 / e 0 .
A i ( z , t ) = Q i ( t ) exp ( i Θ z ) .
Q 0 ( t ) = η   sech ( ω t ) 1 + i A ,
( e 0 , τ , C , β , γ 0 , γ 1 , γ 2 ) = ( 1 , 0.1 , 5 , 8 , 0 , 0 , 10 ) .
A x = λ x + 2 g pert ( 1 + τ 2 t 2 ) [ a 0 b 0 ] d t [ a 0 ( t ) b 0 ( t ) ] [ R 0 ( t ) I 0 ( t ) ] ,
A = [ F G ( a 0 , b 0 ) 0 C 0 0 G + ( b 0 , a 0 ) F + C 0 0 0 0 C γ 1 Θ 0 0 C C 0 Θ 0 γ 1 C 0 0 0 0 C γ 2 Θ 0 0 0 C 0 Θ 0 γ 2 ] ,
F ± = g sat ( 1 + τ 2 T 2 ) γ 0 ± 2 β a 0 b 0 ,
G ± ( v , w ) = ± ( 3 β v 2 + β w 2 + Θ 0 + 1 2 2 T 2 ) ,
g sat = 2 g 0 1 + 2 η 2 / ( ω e 0 ) ,
g pert = g sat 1 + 2 η 2 / ( ω e 0 ) .

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