Abstract

Solitary waves in materials with a cascaded χ(2):χ(2) nonlinearity are investigated, and the implications of the robustness hypothesis for these solitary waves are discussed. Both temporal and spatial solitary waves are studied. First, the basic equations that describe the χ(2):χ(2) nonlinearity in the presence of dispersion or diffraction are derived in the plane-wave approximation, and we show that these equations reduce to the nonlinear Schrödinger equation in the limit of large phase mismatch and can be considered a Hamiltonian deformation of the nonlinear Schrödinger equation. We then proceed to a comprehensive description of all the solitary-wave solutions of the basic equations that can be expressed as a simple sum of a constant term, a term proportional to a power of the hyperbolic secant, and a term proportional to a power of the hyperbolic secant multiplied by the hyperbolic tangent. This formulation includes all the previously known solitary-wave solutions and some exotic new ones as well. Our solutions are derived in the presence of an arbitrary group-velocity difference between the two harmonics, but a transformation that relates our solutions to zero-velocity solutions is derived. We find that all the solitary-wave solutions are zero-parameter and one-parameter families, as opposed to nonlinear-Schrödinger-equation solitons, which are a two-parameter family of solutions. Finally, we discuss the prediction of the robustness hypothesis that there should be a two-parameter family of solutions with solitonlike behavior, and we discuss the experimental requirements for observation of solitonlike behavior.

© 1994 Optical Society of America

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  1. G. I. Stegeman, M. Sheik-Bhae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13–15 (1993).
    [CrossRef] [PubMed]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  3. R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28–30 (1992).
    [CrossRef] [PubMed]
  4. M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, "Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes," Opt. Lett. 18, 1397–1399 (1993).
    [CrossRef] [PubMed]
  5. See, e.g., G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 2.
  6. See, e.g., M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 1.
    [CrossRef]
  7. C. R. Menyuk, "Soliton robustness in optical fibers," J. Opt. Soc. Am. B 10, 1585–1591 (1993).
    [CrossRef]
  8. P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, "Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fiber," Opt. Lett. 11, 464–466 (1986); P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, "Soliton at the zero-group-dispersion wavelength of a single-mode fiber," Opt. Lett. 12, 628–630 (1987).
    [CrossRef] [PubMed]
  9. C. R. Menyuk, "Stability of solitons in birefringent optical fibers, I. Equal propagation amplitudes," Opt. Lett. 12, 614–616 (1987); "Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes," J. Opt. Soc. Am. B 5, 392–402 (1988).
    [CrossRef] [PubMed]
  10. R. J. Hawkins and C. R. Menyuk, "Effect of the detailed Raman cross section on soliton evolution," Opt. Lett. 18, 1999–2001 (1993).
    [CrossRef] [PubMed]
  11. A. Hasegawa and Y. Kodama, "Signal transmission of optical solitons in monomode fiber," Proc. IEEE 69, 1145–1150 (1981).
    [CrossRef]
  12. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  13. C. R. Menyuk, "Origin of solitons in the 'real' world," Phys. Rev. A 33, 4367–4374 (1986); "Application of Lie methods to autonomous Hamiltonian perturbations: second-order calculations," in Nonlinear Evolutions, J. P. P. Léon, ed. (World Scientific, Singapore, 1988), pp. 571–592.
    [CrossRef]
  14. Q. Guo, "Non-linear Schrödinger solitons in media with nonzero second-order nonlinear susceptibility," Quantum Opt. 5, 133–139 (1993).
    [CrossRef]
  15. R. Schiek, "Nonlinear refraction caused by cascaded second-order nonlinearity in optical waveguide structures," J. Opt. Soc. Am. B 10, 1848–1855 (1993).
    [CrossRef]
  16. M. J. Werner and P. D. Drummond, "Simulton solutions for the parametric amplifier," J. Opt. Soc. Am. B 10, 2390–2393 (1993).
    [CrossRef]
  17. M. J. Werner and P. D. Drummond, "Strongly coupled nonlinear parametric solitary waves," Opt. Lett. 19, 613–615 (1994).
    [CrossRef] [PubMed]
  18. A. G. Kalocsai and J. W. Haus, "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity," Phys. Rev. A 49, 574–585 (1994).
    [CrossRef] [PubMed]
  19. M. G. Raymer, P. D. Drummond, and S. J. Carter, "Limits to wideband pulsed squeezing in a traveling-wave parametric amplifier with group-velocity dispersion," Opt. Lett. 16, 1189–1191 (1991).
    [CrossRef] [PubMed]
  20. C. R. Menyuk, "Pulse propagation in an elliptically birefringent Kerr medium," IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  21. R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178–1187 (1984).
    [CrossRef]
  22. J. T. Manassah, "Amplitude and phase of a pulsed second-harmonic signal," J. Opt. Soc. Am. B 4, 1234–1240 (1987).
    [CrossRef]
  23. W. Torruelas, Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Fla. 32826 (personal communication).

1994 (2)

A. G. Kalocsai and J. W. Haus, "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity," Phys. Rev. A 49, 574–585 (1994).
[CrossRef] [PubMed]

M. J. Werner and P. D. Drummond, "Strongly coupled nonlinear parametric solitary waves," Opt. Lett. 19, 613–615 (1994).
[CrossRef] [PubMed]

1993 (7)

1992 (1)

1991 (1)

1989 (1)

C. R. Menyuk, "Pulse propagation in an elliptically birefringent Kerr medium," IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

1987 (2)

1986 (2)

1984 (1)

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178–1187 (1984).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ablowitz, M. J.

See, e.g., M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 1.
[CrossRef]

Agrawal, G. P.

See, e.g., G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 2.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Assanto, G.

Bierlein, J. D.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bosshard, Ch.

Carter, S. J.

Chen, H. H.

DeSalvo, R.

Drummond, P. D.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178–1187 (1984).
[CrossRef]

Gordon, J. P.

Guo, Q.

Q. Guo, "Non-linear Schrödinger solitons in media with nonzero second-order nonlinear susceptibility," Quantum Opt. 5, 133–139 (1993).
[CrossRef]

Hagan, D. J.

Hasegawa, A.

A. Hasegawa and Y. Kodama, "Signal transmission of optical solitons in monomode fiber," Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Haus, J. W.

A. G. Kalocsai and J. W. Haus, "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity," Phys. Rev. A 49, 574–585 (1994).
[CrossRef] [PubMed]

Hawkins, R. J.

Kalocsai, A. G.

A. G. Kalocsai and J. W. Haus, "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity," Phys. Rev. A 49, 574–585 (1994).
[CrossRef] [PubMed]

Kodama, Y.

A. Hasegawa and Y. Kodama, "Signal transmission of optical solitons in monomode fiber," Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

Lee, Y. C.

Manassah, J. T.

J. T. Manassah, "Amplitude and phase of a pulsed second-harmonic signal," J. Opt. Soc. Am. B 4, 1234–1240 (1987).
[CrossRef]

Menyuk, C. R.

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Raymer, M. G.

Reintjes, J.

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178–1187 (1984).
[CrossRef]

Schiek, R.

Segur, H.

See, e.g., M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 1.
[CrossRef]

Sheik-Bahae, M.

Sheik-Bhae, M.

Stegeman, G.

Stegeman, G. I.

Sundheimer, M. L.

Torruelas, W.

W. Torruelas, Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Fla. 32826 (personal communication).

Van Stryland, E.

Van Stryland, E. W.

Vanherzeele, H.

Wai, P. K. A.

Werner, M. J.

IEEE J. Quantum Electron. (2)

C. R. Menyuk, "Pulse propagation in an elliptically birefringent Kerr medium," IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178–1187 (1984).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (9)

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, "Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fiber," Opt. Lett. 11, 464–466 (1986); P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, "Soliton at the zero-group-dispersion wavelength of a single-mode fiber," Opt. Lett. 12, 628–630 (1987).
[CrossRef] [PubMed]

J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

C. R. Menyuk, "Stability of solitons in birefringent optical fibers, I. Equal propagation amplitudes," Opt. Lett. 12, 614–616 (1987); "Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes," J. Opt. Soc. Am. B 5, 392–402 (1988).
[CrossRef] [PubMed]

M. G. Raymer, P. D. Drummond, and S. J. Carter, "Limits to wideband pulsed squeezing in a traveling-wave parametric amplifier with group-velocity dispersion," Opt. Lett. 16, 1189–1191 (1991).
[CrossRef] [PubMed]

R. DeSalvo, D. J. Hagan, M. Sheik-Bahae, G. Stegeman, and H. Vanherzeele, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28–30 (1992).
[CrossRef] [PubMed]

G. I. Stegeman, M. Sheik-Bhae, E. Van Stryland, and G. Assanto, "Large nonlinear phase shifts in second-order nonlinear-optical processes," Opt. Lett. 18, 13–15 (1993).
[CrossRef] [PubMed]

M. L. Sundheimer, Ch. Bosshard, E. W. Van Stryland, G. I. Stegeman, and J. D. Bierlein, "Large nonlinear phase modulation in quasi-phase-matched KTP waveguides as a result of cascaded second-order processes," Opt. Lett. 18, 1397–1399 (1993).
[CrossRef] [PubMed]

R. J. Hawkins and C. R. Menyuk, "Effect of the detailed Raman cross section on soliton evolution," Opt. Lett. 18, 1999–2001 (1993).
[CrossRef] [PubMed]

M. J. Werner and P. D. Drummond, "Strongly coupled nonlinear parametric solitary waves," Opt. Lett. 19, 613–615 (1994).
[CrossRef] [PubMed]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A (1)

A. G. Kalocsai and J. W. Haus, "Nonlinear Schrödinger equation for optical media with quadratic nonlinearity," Phys. Rev. A 49, 574–585 (1994).
[CrossRef] [PubMed]

Quantum Opt. (1)

Q. Guo, "Non-linear Schrödinger solitons in media with nonzero second-order nonlinear susceptibility," Quantum Opt. 5, 133–139 (1993).
[CrossRef]

Other (5)

See, e.g., G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989), Chap. 2.

See, e.g., M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1981), Chap. 1.
[CrossRef]

A. Hasegawa and Y. Kodama, "Signal transmission of optical solitons in monomode fiber," Proc. IEEE 69, 1145–1150 (1981).
[CrossRef]

C. R. Menyuk, "Origin of solitons in the 'real' world," Phys. Rev. A 33, 4367–4374 (1986); "Application of Lie methods to autonomous Hamiltonian perturbations: second-order calculations," in Nonlinear Evolutions, J. P. P. Léon, ed. (World Scientific, Singapore, 1988), pp. 571–592.
[CrossRef]

W. Torruelas, Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Fla. 32826 (personal communication).

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Equations (77)

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E 1 ( z , t ) = A 1 ( z , t ) exp [ i k 1 ( ω 0 ) z - i ω 0 t ] , E 2 ( z , t ) = A 2 ( z , t ) exp [ i k 2 ( 2 ω 0 ) z - 2 i ω 0 t ] ,
2 E z 2 - 1 c 2 2 D t 2 = 0 ,
E ( z , t ) = E 1 ( z , t ) e ^ 1 + E 1 * ( z , t ) e ^ 1 * + E 2 ( z , t ) e ^ 2 + E 2 * ( z , t ) e ^ 2 *
D ( z , t ) = E ( z , t ) + 4 π - t d t Χ ( 1 ) ( t - t ) · E ( z , t ) + 4 π - t d t - t d t Χ ( 2 ) ( t - t , t - t ) · E ( z , t ) E ( z , t ) ,
D ( z , t ) = D 1 ( z , t ) e ^ 1 + D 1 * ( z , t ) e ^ 1 * + D 2 ( z , t ) e ^ 2 + D 2 * ( z , t ) e ^ 2 * ,
D 1 ( z , t ) = U 1 ( z , t ) exp [ i k 1 ( ω 0 ) z - i ω 0 t ] , D 2 ( z , t ) = U 2 ( z , t ) exp [ i k 2 ( 2 ω 0 ) z - 2 i ω 0 t ] ,
U 1 ( z , t ) = A 1 ( z , t ) + 4 π - t d t [ e ^ 1 * · Χ ( 1 ) ( t - t ) · e ^ 1 ] × exp [ i ω 0 ( t - t ) ] A 1 ( z , t ) + 8 π - t d t - t d t [ e ^ 1 * · Χ ( 2 ) ( t - t , t - t ) · e ^ 1 * e ^ 2 ] exp [ 2 i ω 0 ( t - t ) - i ω 0 ( t - t ) ] × A 1 * ( z , t ) A 2 ( z , t ) × exp { - i [ 2 k 1 ( ω 0 ) - k 2 ( 2 ω 0 ) ] z } , U 2 ( z , t ) = A 2 ( z , t ) + 4 π - t d t [ e ^ 2 * · Χ ( 1 ) ( t - t ) · e ^ 2 ] × exp [ 2 i ω 0 ( t - t ) ] A 2 ( z , t ) + 4 π - t d t - t d t [ e ^ 2 * · Χ ( 2 ) ( t - t , t - t ) · e ^ 1 e ^ 1 ] exp [ i ω 0 ( t - t ) + i ω 0 ( t - t ) ] × A 1 ( z , t ) A 2 ( z , t ) exp { i [ 2 k 1 ( ω 0 ) - k 2 ( 2 ω 0 ) ] z } .
χ ˜ 1 ( 1 ) ( ω ) = - d t [ e ^ 1 * · Χ ( 1 ) · e ^ 1 ] exp ( i ω t ) , χ ˜ 2 ( 1 ) ( ω ) = - d t [ e ^ 2 * · Χ ( 1 ) · e ^ 2 ] exp ( i ω t ) ,
χ ˜ 1 ( 2 ) ( 2 ω , - ω ) = - d t - d t [ e ^ 1 * · Χ ( 2 ) ( t , t ) · e ^ 1 * e ^ 2 ] × exp ( 2 i ω t - i ω t ) , χ ˜ 2 ( 2 ) ( ω , ω ) = - d t - d t [ e ^ 2 * · Χ ( 2 ) ( t , t ) · e ^ 1 e ^ 1 ] × exp ( i ω t + i ω t ) ,
χ ˜ 1 ( 1 ) ( ω ) χ ˜ 1 ( 1 ) ( ω 0 ) + χ ˜ 1 ( 1 ) ω | ω 0 ( ω - ω 0 ) + 1 2 2 χ ˜ 1 ( 1 ) ω 2 | ω 0 ( ω - ω 0 ) 2 , χ ˜ 2 ( 1 ) ( ω ) χ ˜ 2 ( 1 ) ( 2 ω 0 ) + χ ˜ 2 ( 1 ) ω | 2 ω 0 ( ω - 2 ω 0 ) + 1 2 2 χ ˜ 2 ( 1 ) ω 2 | 2 ω 0 ( ω - 2 ω 0 ) 2 ,
χ ˜ 1 ( 2 ) ( 2 ω , - ω ) χ ˜ 1 ( 2 ) ( 2 ω 0 , - ω 0 ) , χ ˜ 1 ( 2 ) ( ω , ω ) χ ˜ 1 ( 2 ) ( ω 0 , ω 0 ) .
U 1 ( z , t ) = ˜ 1 A 1 ( z , t ) + i ˜ 1 A 1 ( z , t ) t - 1 2 ˜ 1 2 A 1 ( z , t ) t 2 + 2 ˜ 1 ( 2 ) A 1 * ( z , t ) A 2 ( z , t ) exp ( - i Δ k z ) , U 2 ( z , t ) = ˜ 2 A 2 ( z , t ) + i ˜ 2 A 2 ( z , t ) t - 1 2 ˜ 2 2 A 2 ( z , t ) t 2 + ˜ 2 ( 2 ) A 1 2 ( z , t ) exp ( i Δ k z ) ,
˜ 1 = 1 + 4 π χ ˜ 1 ( 1 ) ( ω 0 ) ,             ˜ 1 = 4 π χ ˜ 1 ( 1 ) ω | ω 0 , ˜ 1 = 4 π 2 χ ˜ 1 ( 1 ) ω 2 | ω 0 , ˜ 2 = 1 + 4 π χ ˜ 2 ( 1 ) ( 2 ω 0 ) ,             ˜ 2 = 4 π χ ˜ 2 ( 1 ) ω | 2 ω 0 , ˜ 2 = 4 π 2 χ ˜ 2 ( 1 ) ω 2 | 2 ω 0 ,
˜ 1 ( 2 ) = 4 π χ ˜ 1 ( 2 ) ( 2 ω 0 , - ω 0 ) , ˜ 2 ( 2 ) = 4 π χ ˜ 2 ( 2 ) ( ω 0 , ω 0 ) .
- k 1 2 A 1 + 2 i k 1 A 1 z + 2 A 1 z 2 + ω 0 2 c 2 ˜ 1 A 1 + i ( ω 0 2 c 2 ˜ 1 + 2 ω 0 c 2 ˜ 1 ) A 1 t - ( ω 0 2 2 c 2 ˜ 1 + 2 ω 0 c 2 ˜ 1 + 1 c 2 ˜ 1 ) 2 A 1 t 2 + 2 ω 0 2 c 2 ˜ 1 ( 2 ) A 1 * A 2 exp ( - i Δ k z ) = 0.
k 1 2 A 1 | k 1 A 1 z | | 2 A 1 z 2 |
ω 0 2 A 1 | ω 0 A 1 t | | 2 A 1 t 2 | .
k 1 2 - ω 0 2 c 2 ˜ 1 = 0 ,
i A 1 z + i ( ω 0 2 2 k 1 c 2 ˜ 1 + ω 0 k 1 c 2 ˜ 1 ) A 1 t = i A 1 z + i k 1 A 1 t = 0 ,
2 A 1 z 2 = ( k 1 ) 2 2 A 1 t 2 ,
i A 1 z + i k 1 A 1 t - 1 2 k 1 2 A 1 t 2 + K 1 A 1 * A 2 exp ( - i Δ k z ) = 0 ,
k 1 = ω 0 2 2 k 1 c 2 ˜ 1 + 2 ω 0 k 1 c 2 ˜ 1 + 1 k 1 c 2 ˜ 1 - 1 k 1 ( k 1 ) 2 = d 2 k 1 d ω 2 | ω 0
i A 2 z + i k 2 A 2 t - 1 2 k 2 2 A 2 t 2 + K 2 A 1 2 exp ( i Δ k z ) = 0 ,
k 2 = d k 2 d ω | 2 ω 0 ,             k 2 = d 2 k 2 d ω 2 | 2 ω 0 , K 2 = 2 ω 0 2 k 2 c 2 ˜ 2 ( 2 ) .
ξ = k 1 τ 2 z , s = t τ - k 1 τ z , δ = ( k 1 - k 2 ) τ k 1 , α = k 2 k 1 ,             β = Δ k τ 2 k 1 , a 1 = K 1 K 2 1 / 2 τ 2 k 1 A 1 , a 2 = K 1 τ 2 k 1 A 2 , r = sgn ( k 1 ) ,
i a 1 ξ - r 2 2 a 1 s 2 + a 1 * a 2 exp ( - i β ξ ) = 0 , i a 2 ξ - i δ a 2 s 2 - α 2 2 a 2 s 2 + a 1 2 exp ( i β ξ ) = 0.
E 1 ( y , z , t ) = A 1 ( y - ρ 1 z , z ) exp ( i k 1 z + i k y y - i ω 0 t ) , E 2 ( y , z , t ) = A 2 ( y - ρ 2 z , z ) exp ( i k 2 z + 2 i k y y - 2 i ω 0 t ) ,
k 1 2 = ω 0 2 c 2 ˜ 1 ( ω 0 ) ,             k 2 2 = 4 ω 0 2 c 2 ˜ 2 ( 2 ω 0 ) .
2 E z 2 + 2 E y 2 + ω 0 2 c 2 D = 0.
U 1 = ˜ 1 A 1 + 2 ˜ 1 ( 2 ) A 1 * A 2 exp ( - i Δ k z ) ,
2 i k 1 A 1 z - 2 i k 1 ρ 1 A 1 y + 2 A 1 y 2 + 2 ω 0 2 c 2 ˜ 1 ( 2 ) A 1 * A 2 exp ( - i Δ k z ) = 0 ,
k 1 = ρ 1 ,             k 1 = - 1 / k 1 ,
i A 1 z - i k 1 A 1 y - 1 2 k 1 2 A 1 y 2 + K 1 A 1 * A 2 exp ( - i Δ k z ) = 0.
i A 2 z - i k 2 A 2 y - 1 2 k 2 2 A 2 y 2 + K 2 A 1 2 exp ( i Δ k z ) = 0 ,
ξ = k 1 η 2 z , s = y η + k 1 η z , δ = - ( k 1 - k 2 ) η k 1 , α = k 2 k 1 ,             β = Δ k η 2 k 1 , a 1 = K 1 K 2 1 / 2 η 2 k 1 A 1 , a 2 = K 1 η 2 k 1 A 2 , r = - 1 ,
i a ^ 1 ξ - r 2 2 a ^ 1 s 2 + a ^ 1 * a ^ 2 = 0 ,
i a ^ 2 ξ - β a ^ 2 - i δ a ^ 2 s - α 2 2 a ^ 2 s 2 + β a ^ 1 2 = 0.
a ^ 2 = β β a ^ 1 2 .
a ^ 2 = β β a ^ 1 2 + 1 β ( - 2 i δ a ^ 1 a ^ 1 s + r a ^ 1 2 a ^ 1 s 2 - α 2 2 a ^ 1 2 s 2 - 2 β β a ^ 1 2 a ^ 1 2 ) .
i a ^ 1 ξ - r 2 2 a ^ 1 s 2 + β β a ^ 1 2 a ^ 1 = O ( 1 / β ) ,
H = - d s ( - β 2 β a ^ 2 a ^ 2 * - i δ 4 β a ^ 2 * a ^ 2 s + i δ 4 β a ^ 2 a ^ 2 * s + r 2 a ^ 1 s a ^ 1 * s + α 4 β a ^ 2 s a ^ 2 * s + 1 2 a ^ 1 2 a ^ 2 * + 1 2 a ^ 1 * 2 a ^ 2 ) ,
[ F , G ] = i - d s ( δ F δ a ^ 1 δ G δ a ^ 1 * - δ F δ a ^ 1 * δ G δ a ^ 1 + 2 β δ F δ a ^ 2 δ G δ a ^ 2 * - 2 β δ F δ a ^ 2 * δ G δ a ^ 2 ) .
a ^ 1 = A sech [ A ( s - r ω ξ - s 0 ) ] × exp [ - i ( r / 2 ) ( A 2 - ω 2 ) ξ - i ω s + i ϕ 0 ] .
a 1 = [ A 1 + i B 1 sech l 1 ( w s - v ξ ) tanh ( w s - v ξ ) + C 1 sech m 1 ( w s - v ξ ) ] exp ( i κ 1 ξ - i ω 1 s ) , a 2 = [ A 2 + i B 2 sech l 2 ( w s - v ξ ) tanh ( w s - v ξ ) + C 2 sech m 2 ( w s - v ξ ) ] exp ( i κ 2 ξ - i ω 2 s ) .
a 1 = j = 0 m 1 [ A 1 , j sech j ( w s - v ξ ) + B 1 , j sech j ( w s - v ξ ) × tanh ( w s - v ξ ) ] exp ( i κ 1 ξ - i ω 1 s ) , a 2 = j = 0 m 2 [ A 2 , j sech j ( w s - v ξ ) + B 2 , j sech j ( w s - v ξ ) × tanh ( w s - v ξ ) ] exp ( i κ 2 ξ - i ω 2 s ) .
ω 1 = - r δ ,             ω 2 = - 2 r δ ,             v = - δ w ,
l 1 = l 2 = 0
m 1 + 2 = m 1 + m 2 ,             2 = 2 m 1 ,
m 1 = 1 ,             m 2 = 2.
a 1 = [ A 1 + i B 1 tanh ( w s - v ξ ) + C 1 sech ( w s - v ξ ) ] × exp ( i κ 1 ξ - i ω 1 s ) , a 2 = [ A 2 + i B 2 tanh ( w s - v ξ ) + C 2 sech 2 ( w s - v ξ ) ] × exp ( i κ 2 ξ - i ω 2 s ) .
ω 2 = 2 ω 1 ,             κ 2 = 2 κ 1 + β .
A 1 C 1 = 0 ,             B 1 C 1 = 0 ;
C 1 = [ w 2 ( δ 2 + w 2 - r β ) ] 1 / 2 ,             C 2 = - r w 2 , κ 1 = ( r / 2 ) ( δ 2 - w 2 ) .
r β < δ 2 + w 2 ,
a 1 = [ A 1 + i B 1 tanh ( w s - v ξ ) ] exp ( i κ 1 ξ - i ω 1 s ) , a 2 = [ A 2 + i B 2 tanh ( w s - v ξ ) + C 2 sech 2 ( w s - v ξ ) × exp ( i κ 2 ξ - i ω 2 s ) ;
A 2 = A 1 2 - B 1 2 κ 2 + δ ω 2 ,             B 2 = 2 A 1 B 1 κ 2 + δ ω 2 ,
[ ( r 2 ω 1 2 - κ 1 ) ( δ ω 2 + κ 2 ) + 2 B 1 2 ] B 1 = 0 , [ ( r 2 ω 1 2 - κ 1 ) ( δ ω 2 + κ 2 ) + 2 A 1 2 ] A 1 = 0.
B 2 = ± 2 B 1 2 κ 2 + δ ω 2 = ± 2 C 2 ,
B 1 = [ w 2 ( 2 w 2 - δ 2 + r β ) ] 1 / 2 ,             A 2 = - C 2 = - r w 2 , κ 1 = ( r / 2 ) ( δ 2 + 2 w 2 ) .
r β > 2 w 2 - δ 2 .
B 2 = ± 2 C 2 = ± 2 r w 2 , A 1 = ± B 1 = ± [ w 2 ( 5 w 2 - δ 2 + r β ) ] 1 / 2 , κ 1 = r 2 δ 2 ± δ w + 5 r 2 w 2 ,
l 1 + 2 = max ( l 1 + m 2 , l 2 + m 1 ) ,             l 2 = l 1 + m 1 , m 1 + 2 = max ( m 1 + m 2 , l 1 + l 2 + 2 ) , m 2 = max ( 2 m 1 , 2 l 1 + 2 ) ,
m 1 = m 2 = 2 , l 1 = l 2 = 0 ; m 1 = m 2 = 2 , l 1 = l 2 = 1 ; m 1 = 1 , m 2 = 2 , l 1 = 1 , l 2 = 0.
a 1 = C 1 sech 2 ( w s - v ξ ) exp ( i κ 1 ξ - i ω 1 s ) , a 2 = C 2 sech 2 ( w s - v ξ ) exp ( i κ 2 ξ - i ω 2 s ) ,
ω 1 = δ 2 α - r , v = r δ w 2 α - r , C 1 = 3 ( α r ) 1 / 2 w 2 , C 2 = - 3 r w 2 , κ 1 = ( r / 2 ) ω 1 2 - 2 r w 2 ,
w 2 = 1 4 r - 2 α ( β + δ 2 2 α - r ) .
v = r ω 1 ( r β / 3 ) 1 / 2 , C 1 = r β ( 1 / 2 ) 1 / 2 , C 2 = - β , w 2 = r β / 3 , κ 1 = r ω 1 2 / 2 - 2 β / 3 ,
ω 1 = r δ / 3 , v = δ w / 3 , C 1 = 3 2 w 2 , C 2 = - 3 r w 2 , κ 1 = r 18 δ 2 - 2 r w 2 ,
ω 1 = δ 2 α - r , v = r δ w 2 α - r , C 1 = 3 ( α r ) 1 / 2 w 2 , C 2 = - 3 r w 2 , A 1 = - 2 ( α r ) 1 / 2 w 2 , A 2 = 2 r w 2 , κ 1 = r 2 δ 2 ( 2 α - r ) 2 + 2 r w 2 ,
w 2 = 1 2 α - 4 r ( β + δ 2 2 α - r ) .
ω 1 = δ 2 α - r , v = r δ w 2 α - r , B 1 = 3 ( - α r ) 1 / 2 w 2 , C 2 = 3 r w 2 , κ 1 = r 2 ω 1 2 - r 2 w 2 ,
w 2 = 1 α + r ( β + δ 2 2 α - r )
a 1 = f 1 ( w s - v ξ ) exp ( i κ 1 ξ - i ω 1 s ) , a 2 = f 2 ( w s - v ξ ) exp ( i κ 2 ξ - i ω 2 s ) ,
ω 1 = δ 2 α - r ,             ω 2 = δ 2 α - r ,             v = r δ w 2 α - r ,
s = s - r δ 2 α - r ξ , ξ = ξ , β = β + δ 2 2 α - r , δ = 0 ,
κ 1 = κ 1 - r 2 δ 2 ( 2 α - r ) 2 ,             κ 2 = κ 2 + 2 ( α - r ) δ 2 ( 2 α - r ) 2 ,
a 1 = f 1 ( w s ) exp ( i κ 1 ξ ) ,             a 2 = f 2 ( w s ) exp ( i κ 2 ξ ) .

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