Abstract

We derive equations modeling the resonant interaction of electric and magnetic components of light fields with metal nanostructures. This paired resonance was recently shown to produce negative refractive index. The model equations are a generalization of the well-known Maxwell–Lorentz model. We demonstrate that in the case of nonlinear polarization and linear magnetization, these equations are equivalent to a system of equations describing the resonant interaction of light with plasmonic oscillations in metal nanospheres. A family of solitary wave solutions is found that is similar to pulses associated with self-induced transparency in the framework of the Maxwell–Bloch model. The evolution of incident optical pulses is studied numerically, as are the collision dynamics of the solitary waves. These simulations reveal that the collision dynamics vary from near perfectly elastic to highly radiative, depending on the relative phase of the initial pulses.

© 2006 Optical Society of America

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  1. J. B. Pendry, "Negative refraction," Contemp. Phys. 45, 191-202 (2004).
    [Crossref]
  2. G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley, 2005).
    [Crossref]
  3. V. M. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
    [Crossref]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [Crossref] [PubMed]
  5. A. K. Iyer, P. C. Kremer, and G. V. Eleftheriades, "Experimental and theoretical verification of focusing in a large, periodically loaded transmission line negative refractive index metamaterial," Opt. Express 11, 696-708 (2003).
    [Crossref] [PubMed]
  6. K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peaks in metamaterials," Opt. Lett. 29, 2623-2625 (2004).
    [Crossref] [PubMed]
  7. H. A. Lorentz, The Theory of Electrons (Dover, 1952).
  8. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).
  9. S. G. Rautian, "Nonlinear saturation spectroscopy of the degenerate electron gas in spherical metallic particles," JETP 85, 451-461 (1997).
    [Crossref]
  10. V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
    [Crossref]
  11. F. Hache, D. Ricard, and C. Flytzanis, "Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects," J. Opt. Soc. Am. B 3, 1647-1655 (1986).
    [Crossref]
  12. A. M. Basharov, and A. I. Maimistov, "Propagation of ultrashort electromagnetic pulses in a Kerr medium with impurity atoms under quasi-resonance conditions," Quantum Electron. 30, 1014-1018 (2000).
    [Crossref]
  13. V. M. Shalaev, ed., Optical Properties of Random Nanostructures, Vol. 82 of Springer-Verlag Topics in Applied Physics (Springer-Verlag, 2002).
    [Crossref]
  14. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
    [Crossref]
  15. S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
    [Crossref]
  16. V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
    [Crossref]
  17. C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
    [Crossref] [PubMed]

2005 (1)

2004 (3)

J. B. Pendry, "Negative refraction," Contemp. Phys. 45, 191-202 (2004).
[Crossref]

K. Aydin, K. Guven, M. Kafesaki, L. Zhang, C. M. Soukoulis, and E. Ozbay, "Experimental observation of true left-handed transmission peaks in metamaterials," Opt. Lett. 29, 2623-2625 (2004).
[Crossref] [PubMed]

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

2003 (1)

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[Crossref] [PubMed]

2000 (2)

A. M. Basharov, and A. I. Maimistov, "Propagation of ultrashort electromagnetic pulses in a Kerr medium with impurity atoms under quasi-resonance conditions," Quantum Electron. 30, 1014-1018 (2000).
[Crossref]

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

1999 (1)

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

1997 (1)

S. G. Rautian, "Nonlinear saturation spectroscopy of the degenerate electron gas in spherical metallic particles," JETP 85, 451-461 (1997).
[Crossref]

1986 (1)

1968 (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[Crossref]

1967 (1)

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[Crossref]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Aydin, K.

Balmain, K. G.

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley, 2005).
[Crossref]

Basharov, A. M.

A. M. Basharov, and A. I. Maimistov, "Propagation of ultrashort electromagnetic pulses in a Kerr medium with impurity atoms under quasi-resonance conditions," Quantum Electron. 30, 1014-1018 (2000).
[Crossref]

Bigot, J.-Y.

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Broyer, M.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Buin, A. K.

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

Cai, W.

Chettiar, U.

Christofilos, D.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Cottancin, E.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Del Fatti, N.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Drachev, V. P.

V. M. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
[Crossref]

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Eleftheriades, G. V.

Flytzanis, C.

Guille, J.

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Guven, K.

Hache, F.

Hahn, E. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[Crossref]

Halt, V.

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Iyer, A. K.

Kafesaki, M.

Kildishev, A. V.

Kremer, P. C.

Lerm, J.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (Dover, 1952).

Maimistov, A. I.

A. M. Basharov, and A. I. Maimistov, "Propagation of ultrashort electromagnetic pulses in a Kerr medium with impurity atoms under quasi-resonance conditions," Quantum Electron. 30, 1014-1018 (2000).
[Crossref]

McCall, S. L.

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[Crossref]

Merle, J.-C.

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Nakotte, H.

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

Ozbay, E.

Pellarin, M.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, "Negative refraction," Contemp. Phys. 45, 191-202 (2004).
[Crossref]

Perakis, I.

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Prvel, B.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Rautian, S. G.

S. G. Rautian, "Nonlinear saturation spectroscopy of the degenerate electron gas in spherical metallic particles," JETP 85, 451-461 (1997).
[Crossref]

Ricard, D.

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[Crossref] [PubMed]

Shalaev, V. M.

V. M. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005).
[Crossref]

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

V. M. Shalaev, ed., Optical Properties of Random Nanostructures, Vol. 82 of Springer-Verlag Topics in Applied Physics (Springer-Verlag, 2002).
[Crossref]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[Crossref] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[Crossref] [PubMed]

Soukoulis, C. M.

Valle, F.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[Crossref]

Voisin, C.

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

Yuan, H.-K.

Zhang, L.

Contemp. Phys. (1)

J. B. Pendry, "Negative refraction," Contemp. Phys. 45, 191-202 (2004).
[Crossref]

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

JETP (1)

S. G. Rautian, "Nonlinear saturation spectroscopy of the degenerate electron gas in spherical metallic particles," JETP 85, 451-461 (1997).
[Crossref]

Nano Lett. (1)

V. P. Drachev, A. K. Buin, H. Nakotte, and V. M. Shalaev, "Size dependent chi3 for conduction electrons in Ag nanoparticles," Nano Lett. 4, 1535-1539 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (1)

V. Halt, J. Guille, J.-C. Merle, I. Perakis, and J.-Y. Bigot, "Electron dynamics in silver nanoparticles: comparison between thin films and glass embedded nanoparticles," Phys. Rev. B 60, 11738-11746 (1999).
[Crossref]

Phys. Rev. Lett. (2)

C. Voisin, D. Christofilos, N. Del Fatti, F. Valle, B. Prvel, E. Cottancin, J. Lerm, M. Pellarin, and M. Broyer, "Size-dependent electron-electron interactions in metal nanoparticles," Phys. Rev. Lett. 85, 2200-2203 (2000).
[Crossref] [PubMed]

S. L. McCall and E. L. Hahn, "Self-induced transparency by pulsed coherent light," Phys. Rev. Lett. 18, 908-911 (1967).
[Crossref]

Quantum Electron. (1)

A. M. Basharov, and A. I. Maimistov, "Propagation of ultrashort electromagnetic pulses in a Kerr medium with impurity atoms under quasi-resonance conditions," Quantum Electron. 30, 1014-1018 (2000).
[Crossref]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
[Crossref] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and µ," Sov. Phys. Usp. 10, 509-514 (1968).
[Crossref]

Other (4)

V. M. Shalaev, ed., Optical Properties of Random Nanostructures, Vol. 82 of Springer-Verlag Topics in Applied Physics (Springer-Verlag, 2002).
[Crossref]

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials: Fundamental Principles and Applications (Wiley, 2005).
[Crossref]

H. A. Lorentz, The Theory of Electrons (Dover, 1952).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

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Figures (5)

Fig. 1
Fig. 1

(a) Electric field amplitude and (b) argument of solitary waves of the Maxwell–Duffing model, plotted for three values of velocity: v = 0.25 (dotted curve), v = 1 (dashed curve), and v = 2 (solid curve).

Fig. 2
Fig. 2

Evolution of electric field amplitude with initial conditions (a) exp ( t 2 2 ) 2 and (b) 2 exp ( t 2 2 ) .

Fig. 3
Fig. 3

(a) Evolution of electric field amplitude with initial condition 5 exp ( t 2 2 ) , (b) output solitary wave amplitude(s) as a function of Gaussian input pulse amplitude A 0 , where the initial condition is given by A 0 exp ( t 2 2 ) .

Fig. 4
Fig. 4

Comparison of the analytical form of the solitary wave solutions (solid curve) with the results of numerical simulations (dashed curve) under conditions identical to those of Fig. 3a.

Fig. 5
Fig. 5

Electric field amplitude showing collision dynamics of solitons for different values of relative phase; (a) Δ ϕ = 0 , (b) Δ ϕ = π .

Equations (48)

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Q ̃ T T + ω r 2 Q ̃ + ϰ Q ̃ 3 = ( e m ) E ̃ .
i Q T + ( ω r ω 0 ) Q + ( 3 ϰ 2 ω 0 ) Q 2 Q = ( e 2 m ω 0 ) E ,
i ( E Z + 1 v g E T ) = 2 π ω 0 N p e c n 0 Q 2 π ω 0 N a d 2 c n 0 Δ a E 2 π i ω 0 N a d 2 c n 0 Δ a 2 E T ,
E = E 2 m ω 0 3 e 2 3 ϰ exp ( i k s Z ) , Q = Q ω 0 2 3 ϰ exp ( i k s Z ) ,
i E z = Q , i Q t + ω Q + Q 2 Q = E .
× E = c 1 B t , × H = c 1 D t ,
B = H + 4 π M , D = E + 4 π P .
E ̂ z = i ω c 1 μ ̂ ( ω ) H ̂ ( ω ) , H ̂ z = i ω c 1 ϵ ̂ ( ω ) E ̂ ( ω ) ,
E ̂ z z = i ( ω c ) 2 μ ̂ ( ω ) ϵ ̂ ( ω ) E ̂ ( ω ) .
P t t = ω p 2 4 π E κ P 3 ,
M t t + ω T 2 M = β 4 π H t t ,
ϵ ̂ ( ω ) μ ̂ ( ω ) = ( ω 2 ω p 2 ) ( ω T 2 ω 2 + β ω 2 ) ω 2 ( ω T 2 ω 2 ) ,
k 2 = ( ω c ) 2 ϵ ̂ ( ω ) μ ̂ ( ω ) .
ω 2 ( k ) = ω p 2 + ω p 2 ω T 2 ( 1 β ) ω p 2 ω T 2 c 2 k 2 .
ω 2 ( k ) = ω 0 2 β ω T 2 ( 1 β ) 2 ( ω p 2 ω 0 2 ) c 2 k 2 .
E z + c 1 H t = 4 π c 1 M t , E x = 0 ,
H z + c 1 E t = 4 π c 1 P t , H y = 0 .
P t t + ω D 2 P + κ P 3 = ω p 2 4 π E ,
M t t + ω T 2 M = β 4 π H t t .
τ = t T , ξ = z L , q = P P 0 , m = M M 0 , e = E E 0 , h = H H 0 ,
T = ω p 1 , L = c ω p 1 , P 0 = ω p κ , ϵ = κ κ , M 0 = P 0 = E 0 4 π
ϵ ζ + h r = m r , h ζ + e r = q τ ,
q τ τ + γ 2 q + ϵ q 3 = e , m τ τ + μ T 2 m = β h τ τ ,
e ¯ ζ i ω h ¯ = i ω m ¯ , h ¯ ζ i ω e ¯ = i ω q ¯ ,
( γ 2 ω 2 ) q ¯ + ϵ F [ q 3 ] ( ω ) = e ¯ , ( μ T 2 ω 2 ) m ¯ = β ω 2 h ¯ ,
μ ̂ ( ω ) = 1 + β ω 2 μ T 2 ω 2 .
e ¯ ζ ζ + ω 2 μ ̂ ( ω ) e ¯ = ω 2 μ ̂ ( ω ) q ¯ ,
( γ 2 ω 2 ) q ¯ + ϵ F [ q 3 ] ( ω ) = e ¯ .
q ( ζ , τ ) = q ̃ ( ζ , τ ) exp ( i ω 0 τ + i k 0 ζ ) + c.c. ,
e ( ζ , τ ) = e ̃ ( ζ , τ ) exp ( i ω 0 τ + i k 0 ζ ) + c.c .
[ ( k 0 + k ) 2 ( ω 0 + ω ) 2 μ ̂ ( ω 0 + ω ) ] e ̃ ( k , ω ) = ( ω 0 + ω ) 2 μ ̂ ( ω 0 + ω ) q ̃ ( k , ω ) ,
[ ( k 0 + k ) 2 ( ω 0 + ω ) 2 μ ̂ ( ω 0 + ω ) ] h ̃ ( k , ω ) = ( ω 0 + ω ) ( k 0 + k ) q ̃ ( k , ω ) ,
[ γ 2 ( ω 0 + ω ) 2 ] q ̃ ( k , ω ) + ϵ F [ q 3 ] ( k , ω ) = e ̃ ( k , ω ) .
( γ 2 ω 2 ) q ¯ ( k , ω ) = e ¯ ( k , ω ) .
[ k 2 ω 2 μ ̂ ( ω ) ] e ¯ ( k , ω ) = ω 2 μ ̂ ( ω ) ( γ 2 ω 2 ) 1 e ¯ ( k , ω ) ,
[ k 2 ω 2 μ ̂ ( ω ) ] h ¯ ( k , ω ) = ω k ( γ 2 ω 2 ) 1 e ¯ ( k , ω ) .
[ k 2 ω 2 ϵ ̂ ( ω ) μ ̂ ( ω ) ] e ¯ ( k , ω ) = 0 ,
[ ( k 0 + k ) 2 ( ω 0 + ω ) 2 ϵ ̂ host ( ω 0 + ω ) μ ̂ ( ω 0 + ω ) ] e ̃ ( k , ω ) = ( ω 0 + ω ) 2 μ ̂ ( ω 0 + ω ) q ̃ ( k , ω ) ,
[ ( k 0 + k ) 2 ( ω 0 + ω ) 2 ϵ ̂ host ( ω 0 + ω ) μ ̂ ( ω 0 + ω ) ] h ̃ ( k , ω ) = ( ω 0 + ω ) ( k 0 + k ) q ̃ ( k , ω ) ,
[ γ 2 ( ω 0 + ω ) 2 ] q ̃ ( k , ω ) + ϵ F [ q 3 ] ( k , ω ) = e ̃ ( k , ω ) .
k 2 ( ω ) = ( ω 0 + ω ) 2 ϵ ̂ host ( ω 0 + ω ) μ ( ω 0 + ω ) .
2 k 0 [ k ( k ω ) ω = ω 0 ω 1 2 ( 2 k ω 2 + 1 2 k 0 k ω k ω ) ω = ω 0 ω 2 ] = 2 k 0 [ k 1 v g 0 ω 1 2 D 0 ω 2 ] .
2 k 0 [ k 1 v g 0 ω 1 2 D 0 ω 2 ] e ̃ ( k , ω ) = ω 0 2 μ ̂ ( ω 0 ) q ̃ ( k , ω ) ,
2 k 0 [ k 1 v g 0 ω 1 2 D 0 ω 2 ] h ̃ ( k , ω ) = ω 0 k 0 q ̃ ( k , ω ) .
i [ ζ + 1 v g 0 τ i 2 D 0 2 τ 2 ] e ̃ ( ζ , τ ) = ω 0 2 2 k 0 μ ̂ ( ω 0 ) q ̃ ( ζ , τ ) .
i q ̃ τ + ( γ ω 0 ) q ̃ 3 ( ϵ ω 0 ) q ̃ 2 q ̃ = e ̃ ω 0 .
E ( t , z ) = v 3 4 exp [ i φ + i Ω t i K ξ i χ ( ξ ) ] ξ 0 [ cosh ( ξ ξ 0 ) + K ] 1 2 ,
Q ( t , z ) = E ( t , z ) exp [ 2 i χ ( ξ ) ] v ,

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