Abstract

We consider the interaction of counter-propagating waves in a bi-directionally pumped ring microresonator with Kerr nonlinearity. We introduce a hierarchy of the mode expansions and envelope functions evolving on different time scales set by the cavity linewidth and nonlinearity, dispersion, and repetition rate, and provide a detailed derivation of the corresponding hierarchy of the coupled mode and of the Lugiato-Lefever-like equations. An effect of the washout of the repetition rate frequencies from the equations governing the dynamics of the counter-propagating waves is elaborated in details.

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    [Crossref]
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    [Crossref]
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2020 (1)

N. M. Kondratiev and V. E. Lobanov, “Modulational instability and frequency combs in whispering-gallery-mode microresonators with backscattering,” Phys. Rev. A 101(1), 013816 (2020).
[Crossref]

2019 (4)

Y. H. Lai, Y. K. Lu, M. G. Suh, Z. Q. Yuan, and K. Vahala, “Observation of the exceptional-point-enhanced Sagnac effect,” Nature 576(7785), 65–69 (2019).
[Crossref]

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

A. L. Gaeta, M. Lipson, and T. J. Kippenberg, “Photonic-chip-based frequency combs,” Nat. Photonics 13(3), 158–169 (2019).
[Crossref]

W. Weng, R. Bouchand, E. Lucas, and T. J. Kippenberg, “Polychromatic Cherenkov Radiation Induced Group Velocity Symmetry Breaking in Counterpropagating Dissipative Kerr Solitons,” Phys. Rev. Lett. 123(25), 253902 (2019).
[Crossref]

2018 (7)

C. Joshi, A. Klenner, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson, and A. L. Gaeta, “Counter-rotating cavity solitons in a silicon nitride microresonator,” Opt. Lett. 43(3), 547 (2018).
[Crossref]

A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Phys. Rep. 729, 1–81 (2018).
[Crossref]

T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361(6402), eaan8083 (2018).
[Crossref]

C. Milián, Y. V. Kartashov, D. V. Skryabin, and L. Torner, “Cavity solitons in a microring dimer with gain and loss,” Opt. Lett. 43(5), 979–982 (2018).
[Crossref]

A. B. Matsko, W. Liang, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, “Fundamental limitations of sensitivity of whispering gallery mode gyroscopes,” Phys. Lett. A 382(33), 2289–2295 (2018).
[Crossref]

M. T. M. Woodley, J. M. Silver, L. Hill, F. Copie, L. D. Bino, S. Zhang, G. Oppo, and P. Del’Haye, “Universal symmetry-breaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators,” Phys. Rev. A 98(5), 053863 (2018).
[Crossref]

D. C. Cole, A. Gatti, S. B. Papp, F. Prati, and L. Lugiato, “Theory of Kerr frequency combs in Fabry-Perot resonators,” Phys. Rev. A 98(1), 013831 (2018).
[Crossref]

2017 (8)

2016 (2)

G. Lin, S. Diallo, J. M. Dudley, and Y. K. Chembo, “Universal nonlinear scattering in ultra-high Q whispering gallery-mode resonators,” Opt. Express 24(13), 14880–14894 (2016).
[Crossref]

M. Dong and H. G. Winful, “Unified approach to cascaded stimulated Brillouin scattering and frequency-comb generation,” Phys. Rev. A 93(4), 043851 (2016).
[Crossref]

2015 (1)

2014 (2)

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

T. Hansson, D. Modotto, and S. Wabnitz, “On the numerical simulation of Kerr frequency combs using coupled mode equations,” Opt. Commun. 312, 134–136 (2014).
[Crossref]

2010 (1)

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[Crossref]

2002 (1)

2001 (1)

D. V. Skryabin, “Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking,” Phys. Rev. E 64(5), 055601 (2001).
[Crossref]

1996 (2)

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43(5), 1071–1077 (1996).
[Crossref]

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E 54(5), 5707–5725 (1996).
[Crossref]

1995 (1)

D. V. Skryabin, A. G. Vladimirov, and A. M. Radin, “Spontaneous phase symmetry breaking due to cavity detuning in a class-A bidirectional ring laser,” Opt. Commun. 116(1-3), 109–115 (1995).
[Crossref]

1990 (1)

R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990).
[Crossref]

1988 (1)

A. R. Aftabizadeh, Y. K. Huang, and J. Wiener, “Bounded solutions for differential equations with reflection of the argument,” J. Math. Anal. Appl. 135(1), 31–37 (1988).
[Crossref]

1987 (1)

L. A. Lugiato and R. Lefever, “Spatial Dissipative Structures in Passive Optical Systems,” Phys. Rev. Lett. 58(21), 2209–2211 (1987).
[Crossref]

1986 (1)

W. R. Christian and L. Mandel, “Frequency dependence of a ring laser with backscattering,” Phys. Rev. A 34(5), 3932–3939 (1986).
[Crossref]

1984 (1)

K. Nozaki and N. Bekki, “Solitons as attractors of a forced dissipative nonlinear Schrödinger equation,” Phys. Lett. A 102(9), 383–386 (1984).
[Crossref]

1983 (1)

D. W. Mc Laughlin, J. V. Moloney, and A. C. Newell, “Solitary waves as fixed points of infinite-dimensional maps in an optical bistable ring cavity,” Phys. Rev. Lett. 51(2), 75–78 (1983).
[Crossref]

1978 (1)

D. J. Kaup and A. C. Newell, “Theory of nonlinear oscillating dipolar excitations in one-dimensional condensates,” Phys. Rev. B 18(10), 5162–5167 (1978).
[Crossref]

Aftabizadeh, A. R.

A. R. Aftabizadeh, Y. K. Huang, and J. Wiener, “Bounded solutions for differential equations with reflection of the argument,” J. Math. Anal. Appl. 135(1), 31–37 (1988).
[Crossref]

Barashenkov, I. V.

I. V. Barashenkov and Y. S. Smirnov, “Existence and stability chart for the ac-driven, damped nonlinear Schrödinger solitons,” Phys. Rev. E 54(5), 5707–5725 (1996).
[Crossref]

I. V. Barashenkov, M. M. Bogdan, and T. Zhanlav, in Nonlinear World: XX International Workshop on Nonlinear and Turbulent Processes in Physics, V. G. Bariakhtar, ed. (World Scientific, 1990).

Bekki, N.

K. Nozaki and N. Bekki, “Solitons as attractors of a forced dissipative nonlinear Schrödinger equation,” Phys. Lett. A 102(9), 383–386 (1984).
[Crossref]

Bino, L. D.

M. T. M. Woodley, J. M. Silver, L. Hill, F. Copie, L. D. Bino, S. Zhang, G. Oppo, and P. Del’Haye, “Universal symmetry-breaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators,” Phys. Rev. A 98(5), 053863 (2018).
[Crossref]

L. D. Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry Breaking of Counter-Propagating Light in a Nonlinear Resonator,” Sci. Rep. 7(1), 43142 (2017).
[Crossref]

Bogdan, M. M.

I. V. Barashenkov, M. M. Bogdan, and T. Zhanlav, in Nonlinear World: XX International Workshop on Nonlinear and Turbulent Processes in Physics, V. G. Bariakhtar, ed. (World Scientific, 1990).

Bouchand, R.

W. Weng, R. Bouchand, E. Lucas, and T. J. Kippenberg, “Polychromatic Cherenkov Radiation Induced Group Velocity Symmetry Breaking in Counterpropagating Dissipative Kerr Solitons,” Phys. Rev. Lett. 123(25), 253902 (2019).
[Crossref]

Brasch, V.

T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8(2), 145–152 (2014).
[Crossref]

Cao, Q.

Q. Cao, H. Wang, C. Dong, H. Jing, R. Liu, X. Chen, L. Ge, Q. Gong, and Y. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017).
[Crossref]

Centeno Neelen, R.

R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990).
[Crossref]

Chembo, Y. K.

A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Phys. Rep. 729, 1–81 (2018).
[Crossref]

G. Lin, S. Diallo, J. M. Dudley, and Y. K. Chembo, “Universal nonlinear scattering in ultra-high Q whispering gallery-mode resonators,” Opt. Express 24(13), 14880–14894 (2016).
[Crossref]

Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A 82(3), 033801 (2010).
[Crossref]

Chen, X.

Q. Cao, H. Wang, C. Dong, H. Jing, R. Liu, X. Chen, L. Ge, Q. Gong, and Y. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017).
[Crossref]

Chen-Jinnai, A.

Chow, W.

Christian, W. R.

W. R. Christian and L. Mandel, “Frequency dependence of a ring laser with backscattering,” Phys. Rev. A 34(5), 3932–3939 (1986).
[Crossref]

Christodoulides, D.

Christodoulides, D. N.

M. P. Hokmabadi, A. Schumer, D. N. Christodoulides, and M. Khajavikhan, “Non-Hermitian ring laser gyroscopes with enhanced Sagnac sensitivity,” Nature 576(7785), 70–74 (2019).
[Crossref]

Coen, S.

A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Phys. Rep. 729, 1–81 (2018).
[Crossref]

Cole, D. C.

D. C. Cole, A. Gatti, S. B. Papp, F. Prati, and L. Lugiato, “Theory of Kerr frequency combs in Fabry-Perot resonators,” Phys. Rev. A 98(1), 013831 (2018).
[Crossref]

Copie, F.

M. T. M. Woodley, J. M. Silver, L. Hill, F. Copie, L. D. Bino, S. Zhang, G. Oppo, and P. Del’Haye, “Universal symmetry-breaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators,” Phys. Rev. A 98(5), 053863 (2018).
[Crossref]

Dale, E.

Del’Haye, P.

M. T. M. Woodley, J. M. Silver, L. Hill, F. Copie, L. D. Bino, S. Zhang, G. Oppo, and P. Del’Haye, “Universal symmetry-breaking dynamics for the Kerr interaction of counterpropagating light in dielectric ring resonators,” Phys. Rev. A 98(5), 053863 (2018).
[Crossref]

A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Phys. Rep. 729, 1–81 (2018).
[Crossref]

L. D. Bino, J. M. Silver, S. L. Stebbings, and P. Del’Haye, “Symmetry Breaking of Counter-Propagating Light in a Nonlinear Resonator,” Sci. Rep. 7(1), 43142 (2017).
[Crossref]

J. M. Silver, K. T. V. Grattan, and P. Del’Haye, “Critical Dynamics of an Asymmetrically Bidirectionally Pumped Optical Microresonator,” arXiv:1912.08262 (2019).

Diallo, S.

Dong, C.

Q. Cao, H. Wang, C. Dong, H. Jing, R. Liu, X. Chen, L. Ge, Q. Gong, and Y. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017).
[Crossref]

Dong, M.

M. Dong and H. G. Winful, “Unified approach to cascaded stimulated Brillouin scattering and frequency-comb generation,” Phys. Rev. A 93(4), 043851 (2016).
[Crossref]

Dudley, J. M.

Eliel, E. R.

R. J. C. Spreeuw, R. Centeno Neelen, N. J. van Druten, E. R. Eliel, and J. P. Woerdman, “Mode coupling in a He-Ne ring laser with backscattering,” Phys. Rev. A 42(7), 4315–4324 (1990).
[Crossref]

Eliyahu, D.

Erkintalo, M.

A. Pasquazi, M. Peccianti, L. Razzari, D. J. Moss, S. Coen, M. Erkintalo, Y. K. Chembo, T. Hansson, S. Wabnitz, P. Del’Haye, X. Xue, A. M. Weiner, and R. Morandotti, “Micro-combs: A novel generation of optical sources,” Phys. Rep. 729, 1–81 (2018).
[Crossref]

Firth, W. J.

W. J. Firth and A. Lord, “Two-dimensional solitons in a Kerr cavity,” J. Mod. Opt. 43(5), 1071–1077 (1996).
[Crossref]

Fujii, S.

Gaeta, A. L.

A. L. Gaeta, M. Lipson, and T. J. Kippenberg, “Photonic-chip-based frequency combs,” Nat. Photonics 13(3), 158–169 (2019).
[Crossref]

C. Joshi, A. Klenner, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson, and A. L. Gaeta, “Counter-rotating cavity solitons in a silicon nitride microresonator,” Opt. Lett. 43(3), 547 (2018).
[Crossref]

T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361(6402), eaan8083 (2018).
[Crossref]

Gatti, A.

D. C. Cole, A. Gatti, S. B. Papp, F. Prati, and L. Lugiato, “Theory of Kerr frequency combs in Fabry-Perot resonators,” Phys. Rev. A 98(1), 013831 (2018).
[Crossref]

Ge, L.

Q. Cao, H. Wang, C. Dong, H. Jing, R. Liu, X. Chen, L. Ge, Q. Gong, and Y. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017).
[Crossref]

Gong, Q.

Q. Cao, H. Wang, C. Dong, H. Jing, R. Liu, X. Chen, L. Ge, Q. Gong, and Y. Xiao, “Experimental Demonstration of Spontaneous Chirality in a Nonlinear Microresonator,” Phys. Rev. Lett. 118(3), 033901 (2017).
[Crossref]

Gorodetsky, M.

T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. Gorodetsky, “Dissipative Kerr solitons in optical microresonators,” Science 361(6402), eaan8083 (2018).
[Crossref]

Gorodetsky, M. L.

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Equations (57)

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c 2 α α 1 E α 1 c 2 α 1 α 1 E α + t 2 ε ( t t , r , θ , z ) E α ( t , r , θ , z ) d t = t 2 N α .
N α = χ α α 1 α 2 α 3 ( 3 ) E α 1 E α 2 E α 3 ,
E α = j = j m i n j m a x b j F α j e i j θ C j ( t ) + c . c . ,   C j B j + ( t ) e i ω j t + B j ( t ) e i ω j t .
N z = N = χ ( 3 ) E z 3 .
ε ( t , θ , r , z ) = ε i d ( t , r , z ) ( 1 + ε i n ( θ ) ) .
b j 2 = 1 2 ϵ v a c S j n j c ,
E z ( t , θ , r , z ) b j p F j p E , E ( t , θ ) = μ ( B μ + e i j μ θ i ω μ t + B μ e i j μ θ + i ω μ t ) + c . c . ,
δ μ = ω μ Ω ,
ω μ = ω 0 + D 1 μ + 1 2 ! D 2 μ 2 + 1 3 ! D 3 μ 3 + ,
E = e i j p θ i Ω t μ B μ + e i δ μ t + i μ θ + e i j p θ + i Ω t μ B μ e i δ μ t + i μ θ + c . c .
= e i j p θ i Ω t μ Q μ + e i μ θ + e i j p θ + i Ω t μ Q μ e i μ θ + c . c .
= e i j p θ i Ω t Q + + e i j p θ + i Ω t Q + c . c .   .
Q μ + = B μ + e i δ μ t ,   Q μ = B μ e i δ μ t ,
Q ± ( t , θ ) = μ Q μ ± e ± i μ θ .
Q ± ( r ) ( t , θ ) = μ Q μ ± e i μ θ ,
Q ± ( r ) ( t , θ ) = Q ± ( t , 2 π θ ) .
A μ + = B μ + e i δ μ t ,   A μ = B μ e i δ μ t ,
δ μ = δ 0 + 1 2 D 2 μ 2 .
E = ( e i j p θ i Ω t μ A μ + e i μ ( θ D 1 t ) + e i j p θ + i Ω t μ A μ e i μ ( θ + D 1 t ) ) + c . c .   .
A ± = μ A μ ± e ± i μ θ ,   A ± ( r ) = μ A μ ± e i μ θ ,
A ± ( r ) ( t , θ ) = A ± ( t , 2 π θ ) .
t 2 N b j p F j p μ e i j μ θ × ( n μ 2 ω μ 2 ε i n ( θ ) B μ + e i ω μ t 2 i ω μ s μ e i ω μ t t B μ + n μ 2 ω μ 2 ε i n ( θ ) B μ e i ω μ t + 2 i ω μ s μ e i ω μ t t B μ ) + b j p F j p μ e i j μ θ × ( n μ 2 ω μ 2 ε i n ( θ ) B μ + e i ω μ t + 2 i ω μ s μ e i ω μ t t B μ + n μ 2 ω μ 2 ε i n ( θ ) B μ e i ω μ t 2 i ω μ s μ e i ω μ t t B μ ) ,
N ( r , z , θ , t ) = μ N j μ ( r , z , t ) e i j μ θ + c . c . .
N j μ P j μ + e i ω j μ t + P j μ e i ω j μ t .
Γ ^ μ μ = 1 2 ω 0 0 2 π e i ( μ μ ) θ ε i n ( θ ) d θ 2 π ,
R ^ μ μ = 1 2 ω 0 0 2 π e i ( 2 j p + μ + μ ) θ ε i n ( θ ) d θ 2 π .
μ Γ ^ μ μ B μ + e i ω μ t i e i ω μ t t B μ + μ Γ ^ μ μ B μ e i ω μ t + i e i ω μ t t B μ μ R ^ μ μ B μ + e i ω μ t μ R ^ μ μ B μ e i ω μ t = π ω 0 n 0 2 V p b j p 2 t 2 N j μ b j p F j p r d r d z π ω 0 n 0 2 V p b j p 2 ( P j μ + e i ω μ t + P j μ e i ω μ t ) b j p F j p r d r d z .
i t B μ + = μ ( Γ ^ μ μ B μ + + R ^ μ μ B μ ) e i ( ω μ ω μ ) t + π ω 0 n 0 2 V p b j p 2 P j μ + b j p F j p r d r d z ,
i t B μ = μ ( Γ ^ μ μ B μ + R ^ μ μ B μ + ) e i ( ω μ ω μ ) t + π ω 0 n 0 2 V p b j p 2 P j μ b j p F j p r d r d z ,
i t B μ ± i t B μ ± + i 1 2 κ ( B μ ± δ ^ μ , 0 H ± e ± i ( ω μ Ω ) t ) .
| H ± | 2 = η π F W ± ,
i t B μ + = i 1 2 κ ( B μ + δ ^ μ , 0 H + e i δ μ t ) R B μ π ω 0 n 0 2 V p b j p 2 P j μ + b j p F j p r d r d z ,
i t B μ = i 1 2 κ ( B μ δ ^ μ , 0 H e i δ μ t ) R B μ + π ω 0 n 0 2 V p b j p 2 P j μ b j p F j p r d r d z .
N = b j p 3 F j p 3 χ ( 3 ) E 3 ,
E 3 = 3 { e i j p θ i Ω t ( | Q + | 2 + 2 | Q | 2 ) Q + + e i j p θ i Ω t ( | Q | 2 + 2 | Q + | 2 ) Q + } + c . c .   .
P j μ + = 3 b j p 3 F j p 3 χ ( 3 ) e i ( ω j μ Ω ) t 0 2 π ( | Q + | 2 + 2 | Q | 2 ) Q + e i μ θ d θ 2 π ,
P j μ = 3 b j p 3 F j p 3 χ ( 3 ) e i ( ω j μ Ω ) t 0 2 π ( | Q | 2 + 2 | Q + | 2 ) Q e i μ θ d θ 2 π .
i t B μ + = i 1 2 κ ( B μ + δ ^ μ , 0 H + e i δ μ t ) R B μ γ e i δ μ t 0 2 π ( | Q + | 2 + 2 | Q | 2 ) Q + e i μ θ d θ 2 π ,
i t B μ = i 1 2 κ ( B μ δ ^ μ , 0 H e i δ μ t ) R B μ + γ e i δ μ t 0 2 π ( | Q | 2 + 2 | Q + | 2 ) Q e i μ θ d θ 2 π ,
γ = 3 2 ω 0 b j p 2 n 0 2 2 π V p χ ( 3 ) F j p 4 r d r d z .
γ = ω 0 S j p n 0 2 π V p n 2 F j p 4 r d r d z .
γ ω 0 n 2 2 S j p n 0 .
i t Q μ + = δ μ Q μ + i 1 2 κ ( Q μ + δ ^ μ , 0 H + ) R Q μ γ 0 2 π ( | Q + | 2 + 2 | Q | 2 ) Q + e i μ θ d θ 2 π ,
i t Q μ = δ μ Q μ i 1 2 κ ( Q μ δ ^ μ , 0 H ) R Q μ + γ 0 2 π ( | Q | 2 + 2 | Q + | 2 ) Q e i μ θ d θ 2 π ,
i t B μ + + i 1 2 κ ( B μ + δ ^ μ , 0 H ± e i δ μ t ) + R B μ = γ e i δ μ t μ 1 μ 2 μ 3 ( δ ^ μ 1 + μ 2 μ 3 , μ B μ 1 + B μ 2 + B μ 3 + e i ( δ μ 1 δ μ 2 + δ μ 3 ) t + 2 δ ^ μ 1 μ 2 + μ 3 , μ B μ 1 + B μ 2 B μ 3 e i ( δ μ 1 δ μ 2 + δ μ 3 ) t ) =
γ μ 1 μ 2 μ 3 δ ^ μ 1 + μ 2 μ 3 , μ ( B μ 1 + B μ 2 + B μ 3 + e i ( δ μ 1 δ μ 2 + δ μ 3 + δ μ ) t + 2 B μ 1 + B μ 3 B μ 2 e i ( δ μ 1 δ μ 3 + δ μ 2 + δ μ ) t ) =
γ μ 1 μ 2 μ 3 δ ^ μ 1 + μ 2 μ 3 , μ ( B μ 1 + B μ 2 + B μ 3 + e i D 2 2 ( μ 2 μ 1 2 μ 2 2 + μ 3 2 ) t + 2 B μ 1 + B μ 2 B μ 3 e i 2 D 1 ( μ 2 μ 3 ) t e i D 2 2 ( μ 2 μ 1 2 μ 3 2 + μ 2 2 ) t ) .
μ 1 + μ 2 = μ 3 + μ ,
i t B μ + + i 1 2 κ ( B μ + δ ^ μ , 0 H ± e i δ μ t ) + R B μ = γ μ 1 μ 2 μ 3 δ ^ μ 1 + μ 2 μ 3 , μ B μ 1 + B μ 2 + B μ 3 + e i D 2 2 ( μ 2 μ 1 2 μ 2 2 + μ 3 2 ) t 2 γ B μ + μ 2 | B μ 2 | 2 .
i t A μ + δ μ A μ + + i 1 2 κ ( A μ + δ ^ μ , 0 H ± ) + R A μ = γ μ 1 μ 2 μ 3 δ ^ μ 1 + μ 2 μ 3 , μ A μ 1 + A μ 2 + A μ 3 + 2 γ A μ + μ 2 | A μ 2 | 2 ,
i t A μ δ μ A μ + i 1 2 κ ( A μ δ ^ μ , 0 H ± ) + R A μ + = γ μ 1 μ 2 μ 3 δ ^ μ 1 + μ 2 μ 3 , μ A μ 1 A μ 2 A μ 3 2 γ A μ μ 2 | A μ 2 + | 2 .
i t A μ + δ μ A μ + + i 1 2 κ ( A μ + δ ^ μ , 0 H ± ) + R A μ = γ 0 2 π | A + | 2 A + e i μ θ d θ 2 π 2 γ A μ + 0 2 π | A | 2 d θ 2 π ,
i t A μ δ μ A μ + i 1 2 κ ( A μ δ ^ μ , 0 H ± ) + R A μ + = γ 0 2 π | A | 2 A e i μ θ d θ 2 π 2 γ A μ 0 2 π | A + | 2 d θ 2 π .
i t Q + = δ 0 Q + + ( i D 1 θ 1 2 ! D 2 θ 2 + i 1 3 ! D 3 θ 3 + ) Q + R Q ( r ) i 1 2 κ ( Q + H + ) γ ( | Q + | 2 + 2 | Q | 2 ) Q + ,
i t Q = δ 0 Q + ( + i D 1 θ 1 2 ! D 2 θ 2 i 1 3 ! D 3 θ 3 + ) Q R Q + ( r ) i 1 2 κ ( Q H ) γ ( | Q | 2 + 2 | Q + | 2 ) Q .
i t A + = δ 0 A + + ( 1 2 ! D 2 θ 2 + i 1 3 ! D 3 θ 3 + ) A + R A ( r ) γ | A + | 2 A + i 1 2 κ ( A + H + ) 2 γ A + 0 2 π | A ( r ) ( θ ) | 2 d θ 2 π ,
i t A ( r ) = δ 0 A ( r ) + ( 1 2 ! D 2 θ 2 + i 1 3 ! D 3 θ 3 + ) A ( r ) R A + γ | A ( r ) | 2 A ( r ) i 1 2 κ ( A ( r ) H ) 2 γ A ( r ) 0 2 π | A + ( θ ) | 2 d θ 2 π .