Abstract

The compression of a cw into a periodic train of noninteracting solitons by a dispersion-decreasing fiber is investigated with a variational method. To model the evolution from the cw to the soliton train, an elliptic-function-based expression is used as the trial function in the averaged Lagrangian. Both a continuous dispersion variation and a step dispersion variation in the fiber are considered. By use of an optimization method based on the approximate variational equations, the optimal dispersion profile required for achieving maximum pulse compression in a fixed length of fiber is determined. The solutions of the approximate equations are compared with full numerical solutions of the governing nonlinear Schrödinger equation, and good agreement is found.

© 1999 Optical Society of America

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References

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  1. S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
    [CrossRef] [PubMed]
  2. S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
    [CrossRef]
  3. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
    [CrossRef]
  4. E. A. Swanson and S. R. Chinn, “40-GHz pulse train generation using soliton compression of a Mach–Zehnder modulator output,” IEEE Photonics Technol. Lett. 7, 114–116 (1995).
    [CrossRef]
  5. V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
    [CrossRef]
  6. D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
    [CrossRef]
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  8. A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).
  9. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954).
  10. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  11. W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
    [CrossRef]
  12. A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Penn., 1985).
  13. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
    [CrossRef]
  14. W. H. Press, S. A. Tuekolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992).
  15. J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).
  16. K. I. McKinnon, “Convergence of the Nelder–Mead simplex method to a non-stationary point,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 148–158 (1998).
  17. B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
    [CrossRef]
  18. R. Haberman, “The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg–de Vries type,” Stud. Appl. Math. 78, 73–90 (1988).
  19. R. Haberman, “Phase shift modulations for stable, oscillatory, travelling, strongly nonlinear waves,” Stud. Appl. Math. 84, 57–69 (1991).

1998

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

K. I. McKinnon, “Convergence of the Nelder–Mead simplex method to a non-stationary point,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 148–158 (1998).

1995

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

E. A. Swanson and S. R. Chinn, “40-GHz pulse train generation using soliton compression of a Mach–Zehnder modulator output,” IEEE Photonics Technol. Lett. 7, 114–116 (1995).
[CrossRef]

1994

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

1993

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
[CrossRef]

1991

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
[CrossRef]

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

R. Haberman, “Phase shift modulations for stable, oscillatory, travelling, strongly nonlinear waves,” Stud. Appl. Math. 84, 57–69 (1991).

1988

R. Haberman, “The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg–de Vries type,” Stud. Appl. Math. 78, 73–90 (1988).

1978

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

1965

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
[CrossRef]

Bogatyrev, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Bubnov, M. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Chamberlin, R. P.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

Chernikov, S. V.

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
[CrossRef]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
[CrossRef]

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Chinn, S. R.

E. A. Swanson and S. R. Chinn, “40-GHz pulse train generation using soliton compression of a Mach–Zehnder modulator output,” IEEE Photonics Technol. Lett. 7, 114–116 (1995).
[CrossRef]

Devyatykh, G. G.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Dianov, E. M.

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
[CrossRef]

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Dong, L.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

Fornberg, B.

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Gur’yanov, A. N.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Haberman, R.

R. Haberman, “Phase shift modulations for stable, oscillatory, travelling, strongly nonlinear waves,” Stud. Appl. Math. 84, 57–69 (1991).

R. Haberman, “The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg–de Vries type,” Stud. Appl. Math. 78, 73–90 (1988).

Kashyap, R.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
[CrossRef]

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Kurkov, A. S.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

Mamyshev, P. V.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
[CrossRef]

McKinnon, K. I.

K. I. McKinnon, “Convergence of the Nelder–Mead simplex method to a non-stationary point,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 148–158 (1998).

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
[CrossRef]

Miroshnichenko, S. I.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
[CrossRef]

Payne, D. N.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

Prokhorov, A. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

Richardson, D. J.

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

Rumyantsev, S. D.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Semenov, S. L.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Semenov, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[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, 1484–1492 (1995).
[CrossRef]

Swanson, E. A.

E. A. Swanson and S. R. Chinn, “40-GHz pulse train generation using soliton compression of a Mach–Zehnder modulator output,” IEEE Photonics Technol. Lett. 7, 114–116 (1995).
[CrossRef]

Sysoliatin, A. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Taylor, J. R.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
[CrossRef]

Whitham, G. B.

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

Comput. J. (UK)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. (UK) 7, 308–313 (1965).
[CrossRef]

Electron. Lett.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Integrated all optical fiber source and multigigahertz soliton pulse train,” Electron. Lett. 29, 1788–1789 (1993).
[CrossRef]

D. J. Richardson, R. P. Chamberlin, L. Dong, and D. N. Payne, “Experimental demonstration of 100 GHz dark soliton generation and propagation using a dispersion decreasing fiber,” Electron. Lett. 30, 1326–1327 (1994).
[CrossRef]

IEEE J. Quantum Electron.

P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. QE-27, 2347–2355 (1991).
[CrossRef]

IEEE Photonics Technol. Lett.

E. A. Swanson and S. R. Chinn, “40-GHz pulse train generation using soliton compression of a Mach–Zehnder modulator output,” IEEE Photonics Technol. Lett. 7, 114–116 (1995).
[CrossRef]

J. Lightwave Technol.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[CrossRef]

Opt. Lett.

Philos. Trans. R. Soc. London, Ser. A

B. Fornberg and G. B. Whitham, “A numerical and theoretical study of certain nonlinear wave phenomena,” Philos. Trans. R. Soc. London, Ser. A 289, 373–403 (1978).
[CrossRef]

Phys. Rev. E

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

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

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence behavior of the Nelder–Mead simplex algorithm in low dimensions,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 112–147 (1998).

K. I. McKinnon, “Convergence of the Nelder–Mead simplex method to a non-stationary point,” SIAM J. Opt. (Soc. Ind. Appl. Math) 9, 148–158 (1998).

Stud. Appl. Math.

R. Haberman, “The modulated phase shift for weakly dissipated nonlinear oscillatory waves of the Korteweg–de Vries type,” Stud. Appl. Math. 78, 73–90 (1988).

R. Haberman, “Phase shift modulations for stable, oscillatory, travelling, strongly nonlinear waves,” Stud. Appl. Math. 84, 57–69 (1991).

Other

A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Penn., 1985).

W. H. Press, S. A. Tuekolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran: The Art of Scientific Computing (Cambridge University, Cambridge, UK, 1992).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

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

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (3)

Fig. 1
Fig. 1

Optimal compressor of length 7.5. The optimal values of the parameters are given in Table 1. (a) Optimal pulse amplitude a as a function of distance z down the fiber for continuous dispersion variation. Full numerical solution, (i) solid curve; solution of approximate equations, (ii) dashed curve; soliton approximation, (iii) dotted curve. (b) Dispersion variation β for the optimal compressor.

Fig. 2
Fig. 2

Optimal compressor of length 5.0. The optimal values of the parameters are given in Table 2. (a) Optimal pulse amplitude a as a function of distance z down the fiber for continuous dispersion variation. Full numerical solution, solid curve; solution of approximate equations, dashed curve. (b) Dispersion variation β for the optimal compressor.

Fig. 3
Fig. 3

Optimal pulse amplitude a as a function of distance z down the fiber for step dispersion variation. Full numerical solution, solid curve; solution of approximate equations, dashed curve. The optimal dispersion parameters are given in Table 3.

Tables (3)

Tables Icon

Table 1 Optimal Values of Dispersion Parameters for a Compressor of Length 7.5 with Continuous Dispersion Variation

Tables Icon

Table 2 Optimal Values of Dispersion Parameters for a Compressor of Length 5.0 with Continuous Dispersion Variation

Tables Icon

Table 3 Optimal Values of Dispersion Parameters for a Compressor of Length 7.5 with Step Dispersion Variation

Equations (35)

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

i uz+12β(z) 2ut2+σ(z)|u|2u=0.
u(0, t)=A cos(t/W).
L=½i(u*uz-uuz*)-β(z)|ut|2+σ(z)|u|4,
u=a cn twexp(iψ)+ig exp(iψ).
L=-2Kw2KwLdt=2-KwKwLdt
Kw=P=(π/2)W,
12L=2PKsin-1mm(ga-ag)-2PKE-m1Kma2ψ-PK2sin-1mmE-m1Kmm1agm-2Pg2ψ-β K2Pm1a2+β KP(2m1-1)E+2m1K3ma2+σ PK(3m1-1)m1K+2(1-2m1)E3m2a4.
4PKsin-1mmdadz+PaK2Kmm1
-sin-1mm2E-m1Kmm1 dmdz-4Pg dψdz=0,
4 sin-1mmdgdz+1mm1-sin-1mmmg dmdz
+4 E-m1Kma dψdz
=43σ (3m1-1)m1K+2(1-2m1)Em2a3-2β K2P2m1K-(2m1-1)E3ma,
2KE-m1Kma dadz+2g dgdz-E2-m1K22m1m2K2a2 dmdz=0,
Pm1-m1 sin-1mmg dadz
+PKsin-1mm(2E-m1K)-Km1a dgdz
+PKE2-m1K2ma2 dψdz
=β K6PE2+4m1EK-2m1E2-3m1K2ma2-σ P3K2m1EK-3m12K2-(1-2m1)E2m2a4.
2m1EK-3m12K2-(1-2m1)E2m2
=E2+4m1EK-2m1E2-3m1K2m=-3π2m32+O(m2)
ddz-|u|2dt=0,
β ddz-|ut|2dt=σ ddz-|u|4dt,
ddzE-m1KmKa2+g2=0.
u=a sech twexp(iψ)+ig exp(iψ).
L=-Ldt.
ddz(aw)=glπσa2-β2w2,
dgdz=-2a3π(σa2-βw-2),
dψdz=σa2-β2w2,
2β aw-8σa3w dadz-β a2w2+2σa4 dwdz=0.
l=3π2β8a0σ.
a03=2σβ a4w-βσ a2w,
w=PK.
r2=3a08σβ 2a2w-2βσ a0+βσ 3π28a0g2.
af3=2σβmin a4w-βminσ a2w.
f=af-i=1n{Rri+RQ[max(0, ri-r_)]2}+apen log5αaf-4,
α=i=1n|ai-ai-1|,a0=0.

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