Abstract

Lumped perturbations of a soliton, such as gain, filtering, and amplitude and phase modulation, are considered. A new formalism is developed to compute the continuum generated by these perturbations through the use of adjoint functions. An exact expression for the continuum in the far field is obtained.

© 1997 Optical Society of America

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References

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  1. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665 (1986).
    [CrossRef] [PubMed]
  2. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  3. H. A. Haus and A. Mecozzi, “Noise of modelocked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
    [CrossRef]
  4. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91 (1992).
    [CrossRef]
  5. D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
    [CrossRef]
  6. V. I. Karpman, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281 (1977).
  7. D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413 (1978).
    [CrossRef]
  8. V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462 (1979).
    [CrossRef]
  9. V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
    [CrossRef]
  10. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  11. Y. S. Kivshar and B. A. Malomed, “Addendum: dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 63, 211 (1991).
    [CrossRef]
  12. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
    [CrossRef] [PubMed]
  13. M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B 10, 1386 (1993).
    [CrossRef]
  14. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic Theory), 3rd ed. (Pergamon, Oxford, 1977).
  15. D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Phys. Fluids 19, 1621 (1976).
  16. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  17. W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484 (1995).
    [CrossRef]
  18. S. Novikov, Theory of Solitons: the Inverse Scattering Methods (Consultants Bureau, New York, 1984).
  19. R. J. Flesch and S. E. Trullinger, “Green’s functions for nonlinear Klein-Gordon kink perturbation theory,” J. Math. Phys. 28, 1619 (1987).
    [CrossRef]
  20. B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
    [CrossRef]
  21. N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
    [CrossRef]
  22. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806 (1992).
    [CrossRef]
  23. N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
    [CrossRef]
  24. D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515 (1992).
    [CrossRef] [PubMed]
  25. F. Matera, A. Mecozzi, M. Romagnoli, and M. Settembre, “Sideband instability induced by periodic power variation in long-distance fiber links,” Opt. Lett. 18, 1499 (1993).
    [CrossRef] [PubMed]
  26. M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30, 1469 (1994).
    [CrossRef]
  27. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157 (1986).
    [CrossRef]
  28. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443 (1990).
    [CrossRef] [PubMed]
  29. A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
    [CrossRef] [PubMed]
  30. W. Forysiak, F. M. Knox, and N. J. Doran, “Average soliton propagation in periodically amplified systems with stepwise dispersion-profiled fiber,” Opt. Lett. 19, 174 (1994).
    [CrossRef] [PubMed]
  31. W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
    [CrossRef]
  32. B. A. Malomed, “Ideal amplification of an ultrashort soliton in a dispersion-decreasing fiber,” Opt. Lett. 19, 341 (1994).
    [CrossRef] [PubMed]

1995 (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484 (1995).
[CrossRef]

1994 (4)

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30, 1469 (1994).
[CrossRef]

W. Forysiak, F. M. Knox, and N. J. Doran, “Average soliton propagation in periodically amplified systems with stepwise dispersion-profiled fiber,” Opt. Lett. 19, 174 (1994).
[CrossRef] [PubMed]

W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
[CrossRef]

B. A. Malomed, “Ideal amplification of an ultrashort soliton in a dispersion-decreasing fiber,” Opt. Lett. 19, 341 (1994).
[CrossRef] [PubMed]

1993 (3)

1992 (5)

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806 (1992).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
[CrossRef]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515 (1992).
[CrossRef] [PubMed]

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91 (1992).
[CrossRef]

1991 (2)

Y. S. Kivshar and B. A. Malomed, “Addendum: dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 63, 211 (1991).
[CrossRef]

A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

1990 (3)

1989 (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

1987 (2)

R. J. Flesch and S. E. Trullinger, “Green’s functions for nonlinear Klein-Gordon kink perturbation theory,” J. Math. Phys. 28, 1619 (1987).
[CrossRef]

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

1986 (2)

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157 (1986).
[CrossRef]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665 (1986).
[CrossRef] [PubMed]

1981 (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

1979 (1)

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462 (1979).
[CrossRef]

1978 (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

1977 (1)

V. I. Karpman, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281 (1977).

1976 (2)

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Phys. Fluids 19, 1621 (1976).

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Andonovic, I.

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
[CrossRef]

Blow, K. J.

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
[CrossRef]

Chbat, M. W.

Dennis, M. L.

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30, 1469 (1994).
[CrossRef]

Doran, N. J.

W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
[CrossRef]

W. Forysiak, F. M. Knox, and N. J. Doran, “Average soliton propagation in periodically amplified systems with stepwise dispersion-profiled fiber,” Opt. Lett. 19, 174 (1994).
[CrossRef] [PubMed]

Duling, I. N.

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30, 1469 (1994).
[CrossRef]

Flesch, R. J.

R. J. Flesch and S. E. Trullinger, “Green’s functions for nonlinear Klein-Gordon kink perturbation theory,” J. Math. Phys. 28, 1619 (1987).
[CrossRef]

Forysiak, W.

W. Forysiak, F. M. Knox, and N. J. Doran, “Average soliton propagation in periodically amplified systems with stepwise dispersion-profiled fiber,” Opt. Lett. 19, 174 (1994).
[CrossRef] [PubMed]

W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
[CrossRef]

Goldman, M. V.

D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Phys. Fluids 19, 1621 (1976).

Gordon, J. P.

Hasegawa, A.

Haus, H. A.

Islam, M. N.

M. W. Chbat, P. R. Prucnal, M. N. Islam, C. E. Soccolich, and J. P. Gordon, “Long-range interference effects of soliton reshaping in optical fibers,” J. Opt. Soc. Am. B 10, 1386 (1993).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157 (1986).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462 (1979).
[CrossRef]

V. I. Karpman, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281 (1977).

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

Kelly, S. M. J.

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806 (1992).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Addendum: dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 63, 211 (1991).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Knox, F. M.

W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
[CrossRef]

W. Forysiak, F. M. Knox, and N. J. Doran, “Average soliton propagation in periodically amplified systems with stepwise dispersion-profiled fiber,” Opt. Lett. 19, 174 (1994).
[CrossRef] [PubMed]

Kodama, Y.

Lai, Y.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic Theory), 3rd ed. (Pergamon, Oxford, 1977).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic Theory), 3rd ed. (Pergamon, Oxford, 1977).

Malomed, B. A.

B. A. Malomed, “Ideal amplification of an ultrashort soliton in a dispersion-decreasing fiber,” Opt. Lett. 19, 341 (1994).
[CrossRef] [PubMed]

Y. S. Kivshar and B. A. Malomed, “Addendum: dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 63, 211 (1991).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

Matera, F.

Mecozzi, A.

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157 (1986).
[CrossRef]

Newell, A. C.

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

Nicholson, D. R.

D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Phys. Fluids 19, 1621 (1976).

Noske, D. U.

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515 (1992).
[CrossRef] [PubMed]

Novikov, S.

S. Novikov, Theory of Solitons: the Inverse Scattering Methods (Consultants Bureau, New York, 1984).

Pandit, N.

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515 (1992).
[CrossRef] [PubMed]

Prucnal, P. R.

Romagnoli, M.

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Settembre, M.

Smith, N. J.

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
[CrossRef]

Smyth, N. F.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Soccolich, C. E.

Solov’ev, V. V.

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Taylor, J. R.

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515 (1992).
[CrossRef] [PubMed]

Trullinger, S. E.

R. J. Flesch and S. E. Trullinger, “Green’s functions for nonlinear Klein-Gordon kink perturbation theory,” J. Math. Phys. 28, 1619 (1987).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Electron. Lett. (2)

N. Pandit, D. U. Noske, S. M. J. Kelly, and J. R. Taylor, “Characteristic instability of fibre loop soliton lasers,” Electron. Lett. 28, 455 (1992).
[CrossRef]

S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28, 806 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. L. Dennis and I. N. Duling, “Experimental study of sideband generation in femtosecond fiber lasers,” IEEE J. Quantum Electron. 30, 1469 (1994).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157 (1986).
[CrossRef]

H. A. Haus and A. Mecozzi, “Noise of modelocked lasers,” IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

J. Lightwave Technol. (2)

W. Forysiak, F. M. Knox, and N. J. Doran, “Stepwise dispersion profiling of periodically amplified soliton systems,” J. Lightwave Technol. 12, 1330 (1994).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329 (1992).
[CrossRef]

J. Math. Phys. (1)

R. J. Flesch and S. E. Trullinger, “Green’s functions for nonlinear Klein-Gordon kink perturbation theory,” J. Math. Phys. 28, 1619 (1987).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

B. A. Malomed, “Propagation of a soliton in a nonlinear waveguide with dissipation and pumping,” Opt. Commun. 61, 192 (1987).
[CrossRef]

Opt. Lett. (6)

Phys. Fluids (1)

D. R. Nicholson and M. V. Goldman, “Damped nonlinear Schrödinger equation,” Phys. Fluids 19, 1621 (1976).

Phys. Rev. A (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
[CrossRef] [PubMed]

Phys. Scr. (1)

V. I. Karpman, “Soliton evolution in the presence of perturbation,” Phys. Scr. 20, 462 (1979).
[CrossRef]

Physica D (1)

V. I. Karpman and V. V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D 3, 487 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

D. J. Kaup and A. C. Newell, “Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory,” Proc. R. Soc. London Ser. A 361, 413 (1978).
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Rev. Mod. Phys. (2)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Addendum: dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 63, 211 (1991).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

D. J. Kaup, “A perturbation expansion for the Zakharov–Shabat inverse scattering transform,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 31, 121 (1976).
[CrossRef]

Sov. Phys. JETP (1)

V. I. Karpman, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281 (1977).

Other (2)

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic Theory), 3rd ed. (Pergamon, Oxford, 1977).

S. Novikov, Theory of Solitons: the Inverse Scattering Methods (Consultants Bureau, New York, 1984).

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Figures (7)

Fig. 1
Fig. 1

Real part of v(t, Ω, 0) where (a) C=1, (b) C=i.

Fig. 2
Fig. 2

(a) Perturbation Im[Δu(t, 0)] due to excess Kerr effect, where =0.1. (b) Continuum Im[Δuc(t, 0)] generated. (c) Continuum spectrum V(Ω).

Fig. 3
Fig. 3

(a) Perturbation Δu(t, 0) due to lumped gain, where g=0.1. (b) Continuum Δuc(t, 0) generated. (c) Continuum spectrum V(Ω).

Fig. 4
Fig. 4

(a) Perturbation Δu(t, 0) due to an amplitude filter, where Ωg2=10. (b) Continuum Δuc(t, 0) generated. (c) Continuum spectrum V(Ω).

Fig. 5
Fig. 5

(a) Perturbation Im[Δu(t, 0)] due to a phase filter, where Ωg2=10. (b) Continuum Im[Δuc(t, 0)] generated. (c) Continuum spectrum V(Ω).

Fig. 6
Fig. 6

(a) Perturbation Δu(t, 0) due to an amplitude modulator, where MAMΩAM2=0.2. (b) Continuum Δuc(t, 0) generated. (c) Continuum spectrum V(Ω).

Fig. 7
Fig. 7

(a) Perturbation Im[Δu(t, 0)] due to a phase modulator, where MPMΩPM2=0.2. (b) Continuum Im[Δuc(t, 0)] generated. (c) Continuum spectrum V(Ω).

Tables (2)

Tables Icon

Table 1 Projection Functions (t, Ω, 0)

Tables Icon

Table 2 Various Perturbations and Their Corresponding Radiation and Spectra

Equations (68)

Equations on this page are rendered with MathJax. Learn more.

-i zu(t, z)=122t2u(t, z)+|u(t, z)|2u(t, z).
-i zΔu(t, z)=122t2Δu(t, z)+2|us(t, z)|2Δu(t, z)+us2(t, z)Δu*(t, z),
us(t, z)=sech(t)exp(iz/2).
Δu(t, 0)=i sech3(t).
Δuc(t, 0)=i [sech3(t)-sech(t)].
u(t, 0)=(1+g)sech(t).
Δu(t, 0)=g sech(t).
Δu(t, 0)=1Ωg22t2sech(t)=1Ωg2[1-2 sech2(t)]sech(t).
Δu(t, 0)=iΩg22t2sech(t)=iΩg2[1-2 sech2(t)]sech(t).
Δu(t, 0)=2iΔωΩg2tsech(t)=-2iΔωΩg2tanh(t)sech(t)
Δu(t, 0)=-2ΔωΩg2tsech(t)=2ΔωΩg2tanh(t)sech(t)
Δu(t, 0)=-MAM2ωAM2t2us(t, 0)=-MAM2ωAM2t2 sech(t).
Δu(t, 0)=-i MPM2ωPM2t2 sech(t).
Δu(t, z)=[Δw(z)fw(t)+Δθ(z)fθ(t)+Δt(z)ft(t)+Δp(z)fp(t)]exp(iz/2)+Δuc(t, z),
ddzΔt(z)=-Δp(z),
ddzΔθ(z)=-12Δw(z),
fw(t)=[1-t tanh(t)]sech(t),
fθ(t)=-i sech(t),
ft(t)=tanh(t)sech(t),
fp(t)=-i t sech(t).
-i zΔu̲(t, z)=122t2Δu̲(t, z)+2|us(t, z)|2Δu(t, z)-us2(t, z)Δu̲*(t, z)
Re -+dtf̲i*(t)fj(t)=δij.
f̲w(t)=sech(t),
f̲θ(t)=-i[1-t tanh(t)]sech(t),
f̲t(t)=t sech(t),
f̲p(t)=-i tanh(t)sech(t).
Δw(z)=Re -+Δu(z, t)f̲w*(t)dt.
-i zf(t, z)=122t2f(t, z),
v(t, z)=-2t2f(t, z)+2 tanh(t) tf(t, z)-tanh2(t)f(t, z)+us2(t, z)f*(t, z),
f(t, Ω, z)=C exp(-iΩt)exp[-i(Ω2/2)z],
v(t, Ω, z)=C[Ω2-2iΩ tanh(t)-tanh2(t)]×exp(-iΩt)×exp[-i(Ω2/2)z]+C* sech2(t)exp(iΩt)exp[i(Ω2/2)z]exp(iz),
Δuc(t, z)=-+ dΩ2πV(Ω)v(t, Ω, z).
v̲(t, Ω, z)=C̲[Ω2-2iΩ tanh(t)-tanh2(t)]exp(-iΩt)×exp[-i(Ω2/2)z]-C̲* sech2(t)exp(iΩt)×exp[i(Ω2/2)z]exp(iz).
limT Re -T/2+T/2dtv̲*(t, Ω, z)v(t, Ω, z)
=limT CC̲*(Ω2+1)2T0ifΩ=Ωotherwise
=δ(Ω-Ω),
|C̲|=1(Ω2+1)2,
V(Ω)=Re -+dtΔu(t, z=0)v̲*(t, Ω, z=0).
V(Ω)=Re -+dtΔu(t, 0)v̲*(t, Ω, 0)=Re -+dti  sech3(t) 1(Ω2+1)2×{-i[Ω2+2iΩ tanh(t)-tanh2(t)]×exp(iΩt)-i sech2(t)exp(-iΩt)}=π4 sechπ2Ω.
Δuc(t, z=0)=-+ dΩ2πV(Ω){i[Ω2-2iΩ tanh(t)-tanh2(t)]exp(-iΩt)-isech2(t)exp(iΩt)}=i [sech3(t)-sech(t)].
us(t, z)=(1+2g)sech[(1+2g)t]×exp{i[(1+2g)2/2]z}.
Δuc(t, 0)=u(t, 0)-us(t, 0)
=(1+g)sech(t)-(1+2g)sech[(1+2g)t]
-g(1+2g){1-2(1+2g)×tanh[(1+2g)t]}sech[(1+2g)t],
f˜(Ω)=-gπ21/2 sechπ2ΩΩ2+1.
V(Ω)=Re -+ dtg sech(t) 1(Ω2+1)2×{[Ω2+2iΩ tanh(t)-tanh2(t)]×exp(iΩt)-sech2(t)exp(-iΩt)}=-gπΩ2+1sechπ2Ω,
Δuc(t, 0)=-+ dΩ2πV(Ω)v(t, Ω, 0)=-+ dΩ2πV(Ω){[Ω2-2iΩ tanh(t)-tanh2(t)]exp(-iΩt)+sech2(t)exp(iΩt)}=-g sech2(t)[cosh(t)-2t sinh(t)],
-+ 1Ω2+1sechπ2Ωexp(-iΩt)dΩ=2 cosh(t)loge[1+exp(-2|t|)]+2|t|exp(-|t|).
f(t, z)=12πiz-+dt0 exp{[i(t-t0)2]/2z}f(t0, z=0)12πiz-+dt0 exp[-(itt0)/z]f(t0, z=0)×exp[(it2)/2z]12πizF-tzexp[(it2)/2z],
Δuc(t, z)=-+dΩ2πV(Ω)v(t, Ω, z)=-+ dΩ2π{V(Ω)[Ω2-2iΩ tanh(t)-tanh2(t)]exp(-iΩt)exp[-i(Ω2/2)z]+V*(Ω)sech2(t)exp(iΩt)exp[i(Ω2/2)z]×exp(iz)}=-2t2f(t, z)+tanh(t) tf(t, z)-tanh2(t)f(t, z)+sech2(t)f*(t, z),
f(t, 0)=-+ dΩ2πV(Ω)exp(-iΩt).
f(t, z)=12πiz exp[i(t2/2z)]V-tz.
Ω22iΩ-1=(Ωi)2=(Ω2+1)exp(-iψ),
ψ=±2 tan-11Ω.
uc(t, z)=12π1iz exp{i[(t2/2z)+2 tan-1(z/|t|)]} 11+t2z2V-tz.
Δu(t, z)=Δu(t, z1)0atz=z1otherwise,
V(Ω, z1)=Re -+dtΔu(t, z1)v̲*(t, Ω, 0),
Δuc(t, z)=-+ dΩ2πV(Ω, z1)v(t, Ω, z-z1).
Δuc(t, z)=-+ dΩ2π0zdz1V(Ω, z1)v(t, Ω, z-z1).
Δu(t, z)=m=0M-1g sech(t)exp(iz/2)δ(z-mza).
Δuc[t, (M-1)za]
=-+ dΩ2π-gπΩ2+1sechπ2Ω(v(t, Ω, 0)×exp{i[(M-1)za/2]}+v(t, Ω,za)×exp{i[(M-2)za/2]}++×v[t, Ω, (M-1)za])=-+ dΩ2π-gπΩ2+1sechπ2Ω×[[Ω2-2iΩ tanh(t)-tanh2(t)]×exp(-iΩt)(exp{i[(M-1)za/2]}+exp[-i(Ω2/2)za]exp{i[(M-2)za/2]}++×exp[-i(Ω2/2)(M-1)za])+sech2(t)×exp(iΩt)(exp{i[(M-1)za/2]}+exp{i[(Ω2/2)+1]za}exp{i[(M-2)za/2]}++exp{i[(Ω2/2)+1](M-1)za})]
=-+ dΩ2π-gπΩ2+1sechπ2Ω×exp[-i(Ω2/2)(M-1)za]sinΩ2+14MzasinΩ2+14za×{[Ω2-2iΩ tanh(t)-tanh2(t)]×exp(-iΩt)-sech2(t)exp(iΩt)}.
sinΩ2+14MzasinΩ2+14za
Ωn2=2nk-1=4nπza-1,
limη+ sin(ηx)sin(x)=πδ(x),
Δuc,n[t, (M-1)za]
=(-1)n+1gπ21Ωn2+1sechπ2Ωn×exp[-i(Ωn2/2)(M-1)za]{[Ωn2-2iΩn tanh(t)-tanh2(t)]exp(-iΩnt)-sech2(t)exp(iΩnt)}.

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