Abstract

The formation of single-soliton or bound-multisoliton states from a single linearly chirped Gaussian pulse in quasi-lossless and lossy fiber spans is examined. The conversion of an input-chirped pulse into soliton states is carried out by virtue of the so-called direct Zakharov–Shabat spectral problem, the solution of which allows one to single out the radiative (dispersive) and soliton constituents of the beam and determine the parameters of the emerging bound state(s). We describe here how the emerging pulse characteristics (the number of bound solitons, the relative soliton power) depend on the input pulse chirp and amplitude.

© 2007 Optical Society of America

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  1. V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional self-modulations of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).
  2. A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
    [CrossRef]
  3. G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).
  4. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).
  5. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).
  6. L. F. Mollenauer and K. Smith, "Demonstration of soliton transmission over more than 4000-km in fiber with loss periodically compensated by Raman gain," Opt. Lett. 13, 675-677 (1988).
    [CrossRef] [PubMed]
  7. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
    [CrossRef]
  8. B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
    [CrossRef]
  9. B. Dany, P. Brindel, E. Pincemin, D. Rouvillain, and O. Leclerc, "Recovered efficiency of filter control in dispersion-managed solitons for optical regeneration applications: analysis and experimental validation," Opt. Lett. 25, 793-795 (2000).
    [CrossRef]
  10. L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
    [CrossRef]
  11. J. Ania-Castañón, "Quasi-lossless transmission using second-order Raman amplification and fibre Bragg grating," Opt. Express 12, 4372-4377 (2004).
    [CrossRef] [PubMed]
  12. J. D. Ania-Castañón and S. K. Turitsyn, "Analytical characterization of a quasi-lossless Raman-amplified fibre transmission scheme," in Lasers and Elctro-Optics Europe, Conference Digest for CLEO Europe 2005 (Optical Society of America, 2005), pp. CJ5-5-TUE.
  13. J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
    [CrossRef] [PubMed]
  14. L. V. Hmurcik and D. J. Kaup, "Solitons created by chirped initial profiles in coherent pulse propagation," J. Opt. Soc. Am. 69, 597-604 (1979).
    [CrossRef]
  15. D. Burak and W. Nasalski, "Gaussian beam to spatial soliton formation in Kerr media," Appl. Opt. 33, 6393-6401 (1994).
    [CrossRef] [PubMed]
  16. D. Burak, "Steering of bright-soliton pairs excited by symmetrical and real initial profiles," Phys. Rev. A 52, 4054-4058 (1995).
    [CrossRef] [PubMed]
  17. D. Krylov, L. Leng, K. Bergman, J. C. Bronski, and J. N. Kutz, "Observation of the breakup of a prechirped N-soliton in an optical fiber," Opt. Lett. 24, 1191-1193 (1999).
    [CrossRef]
  18. A. I. Maumistov and Y. M. Sklyarov, "Influence of the regular phase modulation on formation of optical solitons," Kvantovaya Elektron. (Kiev) 14, 796-803 (1987) A. I. Maumistov and Y. M. Sklyarov,[Sov. J. Quantum Electron. 17, 500-504 (1987)].
    [CrossRef]
  19. M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
    [CrossRef]
  20. K. J. Blow and D. Wood, "The evolution of solitons from non-transform limited pulses," Opt. Commun. 58, 349-354 (1986).
    [CrossRef]
  21. D. J. Kaup and B. A. Malomed, "Variational principle for the Zakharov-Shabat equations," Physica D 84, 319-328 (1995).
    [CrossRef]
  22. M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
    [CrossRef]
  23. J. C. Bronski, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem," Physica D 97, 376-397 (1996).
    [CrossRef]
  24. J. C. Bronski and J. N. Kutz, "Numerical simulation of the semiclassical limits of the focusing nonlinear Schrodinger equation," Phys. Lett. A 254, 325-336 (1999).
    [CrossRef]
  25. M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
    [CrossRef]
  26. S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
    [CrossRef]
  27. Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337-340 (1988).
    [CrossRef]
  28. Y. S. Kivshar, Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia (personal communication, 2006).
  29. E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
    [CrossRef]
  30. W. L. Kath and N. F. Smyth, "Soliton evolution and radiation loss for the nonlinear Schrodinger equation," Phys. Rev. E 51, 1484-1492 (1995).
    [CrossRef]
  31. M. Böhm and F. Mitschke, "Soliton-radiation beat analysis," Phys. Rev. E 73, 066615 (2006).
    [CrossRef]
  32. J. Satsuma and N. Yajima, "Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Suppl. Prog. Theor. Phys. 55, 284-306 (1974).
    [CrossRef]
  33. Y. S. Kivshar and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Rev. Mod. Phys. 61, 763-915 (1989).
    [CrossRef]
  34. E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].
  35. K. J. Blow and N. J. Doran, "The asymptotic dispersion of soliton pulses in lossy fibers," Opt. Commun. 52, 367-370 (1985).
    [CrossRef]

2006 (2)

M. Böhm and F. Mitschke, "Soliton-radiation beat analysis," Phys. Rev. E 73, 066615 (2006).
[CrossRef]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

2004 (1)

2002 (2)

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

2000 (1)

1999 (3)

D. Krylov, L. Leng, K. Bergman, J. C. Bronski, and J. N. Kutz, "Observation of the breakup of a prechirped N-soliton in an optical fiber," Opt. Lett. 24, 1191-1193 (1999).
[CrossRef]

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

J. C. Bronski and J. N. Kutz, "Numerical simulation of the semiclassical limits of the focusing nonlinear Schrodinger equation," Phys. Lett. A 254, 325-336 (1999).
[CrossRef]

1998 (1)

S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
[CrossRef]

1996 (2)

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

J. C. Bronski, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem," Physica D 97, 376-397 (1996).
[CrossRef]

1995 (4)

D. J. Kaup and B. A. Malomed, "Variational principle for the Zakharov-Shabat equations," Physica D 84, 319-328 (1995).
[CrossRef]

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
[CrossRef]

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

D. Burak, "Steering of bright-soliton pairs excited by symmetrical and real initial profiles," Phys. Rev. A 52, 4054-4058 (1995).
[CrossRef] [PubMed]

1994 (2)

D. Burak and W. Nasalski, "Gaussian beam to spatial soliton formation in Kerr media," Appl. Opt. 33, 6393-6401 (1994).
[CrossRef] [PubMed]

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
[CrossRef]

1989 (1)

Y. S. Kivshar and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

1988 (2)

1987 (1)

A. I. Maumistov and Y. M. Sklyarov, "Influence of the regular phase modulation on formation of optical solitons," Kvantovaya Elektron. (Kiev) 14, 796-803 (1987) A. I. Maumistov and Y. M. Sklyarov,[Sov. J. Quantum Electron. 17, 500-504 (1987)].
[CrossRef]

1986 (2)

K. J. Blow and D. Wood, "The evolution of solitons from non-transform limited pulses," Opt. Commun. 58, 349-354 (1986).
[CrossRef]

E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].

1985 (1)

K. J. Blow and N. J. Doran, "The asymptotic dispersion of soliton pulses in lossy fibers," Opt. Commun. 52, 367-370 (1985).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

1979 (1)

1974 (1)

J. Satsuma and N. Yajima, "Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Suppl. Prog. Theor. Phys. 55, 284-306 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional self-modulations of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Agarwal, A.

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Anderson, D.

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
[CrossRef]

Ania-Castañón, J.

Ania-Castañón, J. D.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

J. D. Ania-Castañón and S. K. Turitsyn, "Analytical characterization of a quasi-lossless Raman-amplified fibre transmission scheme," in Lasers and Elctro-Optics Europe, Conference Digest for CLEO Europe 2005 (Optical Society of America, 2005), pp. CJ5-5-TUE.

Bergman, K.

Blow, K. J.

K. J. Blow and D. Wood, "The evolution of solitons from non-transform limited pulses," Opt. Commun. 58, 349-354 (1986).
[CrossRef]

K. J. Blow and N. J. Doran, "The asymptotic dispersion of soliton pulses in lossy fibers," Opt. Commun. 52, 367-370 (1985).
[CrossRef]

Böh, M.

M. Böhm and F. Mitschke, "Soliton-radiation beat analysis," Phys. Rev. E 73, 066615 (2006).
[CrossRef]

Brindel, P.

B. Dany, P. Brindel, E. Pincemin, D. Rouvillain, and O. Leclerc, "Recovered efficiency of filter control in dispersion-managed solitons for optical regeneration applications: analysis and experimental validation," Opt. Lett. 25, 793-795 (2000).
[CrossRef]

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

Bronski, J. C.

J. C. Bronski and J. N. Kutz, "Numerical simulation of the semiclassical limits of the focusing nonlinear Schrodinger equation," Phys. Lett. A 254, 325-336 (1999).
[CrossRef]

D. Krylov, L. Leng, K. Bergman, J. C. Bronski, and J. N. Kutz, "Observation of the breakup of a prechirped N-soliton in an optical fiber," Opt. Lett. 24, 1191-1193 (1999).
[CrossRef]

J. C. Bronski, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem," Physica D 97, 376-397 (1996).
[CrossRef]

Burak, D.

D. Burak, "Steering of bright-soliton pairs excited by symmetrical and real initial profiles," Phys. Rev. A 52, 4054-4058 (1995).
[CrossRef] [PubMed]

D. Burak and W. Nasalski, "Gaussian beam to spatial soliton formation in Kerr media," Appl. Opt. 33, 6393-6401 (1994).
[CrossRef] [PubMed]

Burtsev, S.

S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
[CrossRef]

Camassa, R.

S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
[CrossRef]

Chen, X.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

Dany, B.

B. Dany, P. Brindel, E. Pincemin, D. Rouvillain, and O. Leclerc, "Recovered efficiency of filter control in dispersion-managed solitons for optical regeneration applications: analysis and experimental validation," Opt. Lett. 25, 793-795 (2000).
[CrossRef]

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

Desaix, M.

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
[CrossRef]

Desaux, M.

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Desurvire, E.

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

Dianov, E. M.

E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].

Doran, N. J.

K. J. Blow and N. J. Doran, "The asymptotic dispersion of soliton pulses in lossy fibers," Opt. Commun. 52, 367-370 (1985).
[CrossRef]

Ellingham, T. J.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).

Helczynski, L.

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

Hmurcik, L. V.

Iannone, E.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

Ibbotson, R.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

Kath, W. L.

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

Kaup, D. J.

D. J. Kaup and B. A. Malomed, "Variational principle for the Zakharov-Shabat equations," Physica D 84, 319-328 (1995).
[CrossRef]

L. V. Hmurcik and D. J. Kaup, "Solitons created by chirped initial profiles in coherent pulse propagation," J. Opt. Soc. Am. 69, 597-604 (1979).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337-340 (1988).
[CrossRef]

Y. S. Kivshar, Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia (personal communication, 2006).

Kodama, Y.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).

Krylov, D.

Kumar, P.

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

Kutz, J. N.

J. C. Bronski and J. N. Kutz, "Numerical simulation of the semiclassical limits of the focusing nonlinear Schrodinger equation," Phys. Lett. A 254, 325-336 (1999).
[CrossRef]

D. Krylov, L. Leng, K. Bergman, J. C. Bronski, and J. N. Kutz, "Observation of the breakup of a prechirped N-soliton in an optical fiber," Opt. Lett. 24, 1191-1193 (1999).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
[CrossRef]

Leclerc, O.

B. Dany, P. Brindel, E. Pincemin, D. Rouvillain, and O. Leclerc, "Recovered efficiency of filter control in dispersion-managed solitons for optical regeneration applications: analysis and experimental validation," Opt. Lett. 25, 793-795 (2000).
[CrossRef]

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

Leng, L.

Lisak, M.

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
[CrossRef]

Malomed, B. A.

D. J. Kaup and B. A. Malomed, "Variational principle for the Zakharov-Shabat equations," Physica D 84, 319-328 (1995).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

Matera, F.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

Maumistov, A. I.

A. I. Maumistov and Y. M. Sklyarov, "Influence of the regular phase modulation on formation of optical solitons," Kvantovaya Elektron. (Kiev) 14, 796-803 (1987) A. I. Maumistov and Y. M. Sklyarov,[Sov. J. Quantum Electron. 17, 500-504 (1987)].
[CrossRef]

Mecozzi, A.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

Mikhailov, A. V.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
[CrossRef]

Mitschke, F.

M. Böhm and F. Mitschke, "Soliton-radiation beat analysis," Phys. Rev. E 73, 066615 (2006).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer and K. Smith, "Demonstration of soliton transmission over more than 4000-km in fiber with loss periodically compensated by Raman gain," Opt. Lett. 13, 675-677 (1988).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Nasalski, W.

Nikonova, Z. S.

E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].

Pincemin, E.

Quiroga-Teixeiro, M. L.

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

Rouvillain, D.

Satsuma, J.

J. Satsuma and N. Yajima, "Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Suppl. Prog. Theor. Phys. 55, 284-306 (1974).
[CrossRef]

Serkin, V. N.

E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].

Settembre, M.

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional self-modulations of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Shimokhin, I. A.

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
[CrossRef]

Sklyarov, Y. M.

A. I. Maumistov and Y. M. Sklyarov, "Influence of the regular phase modulation on formation of optical solitons," Kvantovaya Elektron. (Kiev) 14, 796-803 (1987) A. I. Maumistov and Y. M. Sklyarov,[Sov. J. Quantum Electron. 17, 500-504 (1987)].
[CrossRef]

Smith, K.

Smyth, N. F.

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

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Su, Y. K.

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

Timofeyev, I.

S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
[CrossRef]

Turitsyn, S. K.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

J. D. Ania-Castañón and S. K. Turitsyn, "Analytical characterization of a quasi-lossless Raman-amplified fibre transmission scheme," in Lasers and Elctro-Optics Europe, Conference Digest for CLEO Europe 2005 (Optical Society of America, 2005), pp. CJ5-5-TUE.

Wang, L. J.

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, "The evolution of solitons from non-transform limited pulses," Opt. Commun. 58, 349-354 (1986).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, "Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Suppl. Prog. Theor. Phys. 55, 284-306 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional self-modulations of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Zhang, L.

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973).
[CrossRef]

Electron. Lett. (1)

B. Dany, P. Brindel, O. Leclerc, and E. Desurvire, "Transoceanic 4 40 Gbit/s system combining dispersion-managed soliton transmission and new 'black-box' in-line optical regeneration," Electron. Lett. 35, 98-100 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. J. Wang, A. Agarwal, Y. K. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

J. Comput. Phys. (1)

S. Burtsev, R. Camassa, and I. Timofeyev, "Numerical algorithms for the direct spectral transform with applications to nonlinear Schrodinger type systems," J. Comput. Phys. 147, 166-186 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. A (2)

Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337-340 (1988).
[CrossRef]

M. Desaix, D. Anderson, M. Lisak, and M. L. Quiroga-Teixeiro, "An approximation procedure for the Zakharov-Shabat scattering problem for real single-humped potentials," J. Phys. A 29, 2493-2498 (1996).
[CrossRef]

Kvantovaya Elektron. (Kiev) (2)

A. I. Maumistov and Y. M. Sklyarov, "Influence of the regular phase modulation on formation of optical solitons," Kvantovaya Elektron. (Kiev) 14, 796-803 (1987) A. I. Maumistov and Y. M. Sklyarov,[Sov. J. Quantum Electron. 17, 500-504 (1987)].
[CrossRef]

E. M. Dianov, Z. S. Nikonova, and V. N. Serkin, "The effect of optical losses on pulse propagation dynamics in single-mode fiber waveguides," Kvantovaya Elektron. (Kiev) 13, 331-337 (1986) E. M. Dianov, Z. S. Nikonova, and V. N. Serkin,[Sov. J. Quantum Electron. 11, 219-222 (1986)].

Opt. Commun. (2)

K. J. Blow and N. J. Doran, "The asymptotic dispersion of soliton pulses in lossy fibers," Opt. Commun. 52, 367-370 (1985).
[CrossRef]

K. J. Blow and D. Wood, "The evolution of solitons from non-transform limited pulses," Opt. Commun. 58, 349-354 (1986).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Lett. A (1)

J. C. Bronski and J. N. Kutz, "Numerical simulation of the semiclassical limits of the focusing nonlinear Schrodinger equation," Phys. Lett. A 254, 325-336 (1999).
[CrossRef]

Phys. Rev. A (1)

D. Burak, "Steering of bright-soliton pairs excited by symmetrical and real initial profiles," Phys. Rev. A 52, 4054-4058 (1995).
[CrossRef] [PubMed]

Phys. Rev. E (4)

M. Desaix, D. Anderson, and M. Lisak, "Variational approach to the Zakharov-Shabat scattering problem," Phys. Rev. E 50, 2253-2256 (1994).
[CrossRef]

M. Desaux, L. Helczynski, D. Anderson, and M. Lisak, "Propagation properties of chirped soliton pulses in optical nonlinear Kerr media," Phys. Rev. E 65, 056602 (2002).
[CrossRef]

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

M. Böhm and F. Mitschke, "Soliton-radiation beat analysis," Phys. Rev. E 73, 066615 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental-observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

J. D. Ania-Castañón, T. J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, and S. K. Turitsyn, "Ultralong Raman fiber lasers as virtually lossless optical media," Phys. Rev. Lett. 96, 023902 (2006).
[CrossRef] [PubMed]

Physica D (3)

E. A. Kuznetsov, A. V. Mikhailov, and I. A. Shimokhin, "Nonlinear interaction of solitons and radiation," Physica D 87, 201-215 (1995).
[CrossRef]

D. J. Kaup and B. A. Malomed, "Variational principle for the Zakharov-Shabat equations," Physica D 84, 319-328 (1995).
[CrossRef]

J. C. Bronski, "Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem," Physica D 97, 376-397 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, "Dynamics of solitons in nearly integrable systems," Rev. Mod. Phys. 61, 763-915 (1989).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of 2-dimensional self-focusing and one-dimensional self-modulations of waves in nonlinear media," Sov. Phys. JETP 34, 62-69 (1972).

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, "Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media," Suppl. Prog. Theor. Phys. 55, 284-306 (1974).
[CrossRef]

Other (5)

Y. S. Kivshar, Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia (personal communication, 2006).

J. D. Ania-Castañón and S. K. Turitsyn, "Analytical characterization of a quasi-lossless Raman-amplified fibre transmission scheme," in Lasers and Elctro-Optics Europe, Conference Digest for CLEO Europe 2005 (Optical Society of America, 2005), pp. CJ5-5-TUE.

G. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, 1998).

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Figures (10)

Fig. 1
Fig. 1

Density plot of the number of solitons [numbers of discrete eigenvalues for the associated ZSSP, Eq. (4)] generated from the input-chirped Gaussian profile, Eq. (9), as a function of the amplitude A and chirp C values. Lossless case.

Fig. 2
Fig. 2

Density plot of the relative power of the solution, N soliton ( N radiation + N soliton ) , see Eq. (7), generated from the input-chirped Gaussian profile, Eq. (9), as a function of the amplitude A and chirp C values. Lossless case.

Fig. 3
Fig. 3

Density plot of the part of the Hamiltonian H soliton , see Eq. (8), transferred into the solitons generated from the input-chirped Gaussian profile, Eq. (9), as a function of the amplitude A and chirp C values. Lossless case.

Fig. 4
Fig. 4

Propagation of a Gaussian pulse for different values of amplitude A and chirp C (see Fig.1 for the corresponding number of bound states calculated for each case). Lossless case, α = 0 was assumed, (a) Here A = 2 , C = 0 , and a two-soliton bound state is formed, (b) A = 2 , C = 2 , and a single soliton is created, (c) For A = 2 , C = 7 , large chirp prevents the creation of solitons, and the power disperses quickly with the propagation.

Fig. 5
Fig. 5

Amplitude threshold for soliton creation versus chirp. Lossless case.

Fig. 6
Fig. 6

Breaking of a bound five-soliton state in a lossy fiber. The initial profile was given by Eq. (9) with A = 5 , C = 0 . Pulse power is also shown as a density-plot in the inset. The propagation distance was L = 1 .

Fig. 7
Fig. 7

Evolution of discrete ZSSP eigenvalues with the propagation distance z, corresponding to the lossy case shown in Fig. 6 and the same parameters of the initial chirped Gaussian pulse. (a) At z = 0 , we have five well-separated eigenvalues located at the imaginary axis, so that the bound state is the so-called breather state and soliton velocities are 0. (b) At z = 0.13 , the lowest eigenvalue is now below the soliton threshold and also a pair of eigenvalues acquires symmetric real parts (velocities). (c) At z = 0.54 , the picture is qualitatively the same as in the case (b), but now the lowest eigenvalue reappears again. (d) At z = 0.73 , one observes a separation of yet another pair of eigenvalues.

Fig. 8
Fig. 8

Density plot of the number of solitons [numbers of discrete eigenvalues for the associated ZSSP, Eq. (4)] generated from the input-chirped Gaussian profile, Eq. (9), as a function of the input amplitude A and chirp C values. Lossy case, α = 1.15 , propagation distance L = 1 .

Fig. 9
Fig. 9

Density plot of the relative power of the solution, N soliton ( N radiation + N soliton ) , generated from the input-chirped Gaussian profile, Eq. (9), as a function of the input amplitude A and chirp C values. Lossy case, α = 1.15 , propagation distance L = 1 .

Fig. 10
Fig. 10

Density plot of the part of the Hamiltonian H transferred into the soliton part generated from the input-chirped Gaussian profile, Eq. (9), as a function of the input amplitude A and chip C values. Lossey case, α = 1.15 , propagation distance L = 1 .

Equations (10)

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i ψ z + i α 2 ψ β 2 2 ψ t t + γ ψ 2 ψ = 0 .
i ψ z + i α 2 ψ 1 2 ψ t t + ψ 2 ψ = 0 .
ψ ( z , t ) = 2 η cosh [ 2 η ( t T 0 + 2 ξ z ) ] e 2 i ξ t + i σ 0 + 2 ( η 2 ξ 2 ) z .
d φ 1 d t = ψ φ 2 i ζ φ 1 ,
d φ 2 d t = ψ * φ 1 + i ζ φ 2 .
N = d t ψ 2 ,
H = 1 2 d t ( ψ t 2 ψ 4 ) .
N = 4 k η k + d ζ n ( ζ ) ,
H = 16 k ( ξ k 2 η k η k 3 3 ) + d ζ h ( ζ ) ,
ψ 0 ( t , 0 ) = A exp [ 1 2 ( 1 + i C ) t 2 ] ,

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