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Solitons Around Us: Integrable, Hamiltonian and Dissipative Systems

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Optical Solitons

Part of the book series: Lecture Notes in Physics ((LNP,volume 613))

Abstract

In recent years, the notion of solitons has been extended to various systems which are not necessarily integrable. We extend the notion of solitons and include a wider range of systems in our treatment. These include dissipative systems, Hamiltonian systems and a particular case of them, viz. integrable systems. We use the broad definition of solitons and give some examples to support this claim of ubiquity.

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References

  1. V. E. Zakharov and A. B. Shabat: Sov. Phys. JETP 34, 62 (1971).

    MathSciNet  ADS  Google Scholar 

  2. A. D. Boardman: Editorial, Photonics Science News 5, 2 (1999).

    Google Scholar 

  3. H. Goldstein: Classical Mechanics, (2-nd ed., Addison-Wesley, New York, 1980).

    MATH  Google Scholar 

  4. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris: Solitons and Nonlinear Wave Equations, (Academic Press, London, 1984).

    Google Scholar 

  5. A. C. Newell: Solitons in Mathematics and Physics, (Society of Industrial and Applied Mathematics, Arizona, 1985).

    Google Scholar 

  6. M. J. Ablowitz and P. A. Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering, (London Mathematical Society Lecture Notes Series 149, Cambridge University Press, Cambridge, 1991).

    Book  MATH  Google Scholar 

  7. R. K. Bullough and P. J. Caudrey: Solitons, (Springer-Verlag, Berlin, 1980).

    MATH  Google Scholar 

  8. S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov: Theory of Solitons-The Inverse Scattering Method, (Plenum, New York, 1984).

    Google Scholar 

  9. N. N. Akhmediev and A. Ankiewicz: Solitons: Nonlinear Pulses and beams, (Chapman & Hall, London, 1997).

    Google Scholar 

  10. V. S. Gerdzhikov and M. I. Ivanov: Teor. Mat. Fiz. 52, 89 (1982) Theor. Math. Phys. 52, 676 (1982).

    MathSciNet  Google Scholar 

  11. K. Nakkeeran, K. Porsezian, P. Shanmugha Sundaram, and A. Mahalingam: Phys. Rev. Lett. 80 1425 (1998).

    Article  ADS  Google Scholar 

  12. V. G. Makhan’kov and O. K. Pashaev: Teor. Mat. Fiz. 53, 55 (1982) Theor. Math. Phys. 53, 979 (1982).

    MathSciNet  Google Scholar 

  13. C. B. Manakov: Sov. Phys. JETP 38, 248 (1974).

    ADS  MathSciNet  Google Scholar 

  14. N. Akhmediev and A. Ankiewicz: Chaos 10, 600 (2000).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. M. Mitchell, Z. Chen, M. Shih and M. Segev: Phys. Rev. Lett. 77, 490 (1996).

    Article  ADS  Google Scholar 

  16. M. Segev and G. Stegeman: Physics Today No 8, 42 (1998).

    Google Scholar 

  17. N. N. Akhmediev and A. Ankiewicz: Photonics Science News 5, 13 (1999).

    Google Scholar 

  18. D. N. Christodoulides, T. H. Coskun and R. I. Joseph: Opt. Lett. 22, 1080 (1997).

    Article  ADS  Google Scholar 

  19. A. Ankiewicz, W. Królikowski, and N. N. Akhmediev: Phys. Rev. E 59, 6079 (1999).

    Article  ADS  Google Scholar 

  20. A. W. Snyder, D. J. Mitchell: Phys. Rev. Lett. 80, 1422 (1998).

    Article  ADS  Google Scholar 

  21. N. N. Akhmediev and J. M. Soto-Crespo: Phys. Rev. E 49, 4519 (1994).

    Article  ADS  Google Scholar 

  22. N. N. Akhmediev and J. M. Soto-Crespo: Phys. Rev. E 49, 5742 (1994)

    Article  ADS  Google Scholar 

  23. V. E. Zakharov: Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 431 (1974).

    Google Scholar 

  24. L. D. Faddeev and L. A. Takhtadjan: Hamiltonian Methods in the Theory of Solitons, (Springer-Verlag, Berlin, 1987).

    MATH  Google Scholar 

  25. V. E. Zakharov and A. M. Rubenchik: Sov. Phys. JETP 38, 494 (1973).

    ADS  Google Scholar 

  26. J. J. Rasmussen and K. Ripdal: Physica Scripta 33, 481 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. N. N. Akhmediev: Optical and Quantum Electronics 30, 535 (1998).

    Article  Google Scholar 

  28. E. A. Ostrovskaya, N. N. Akhmediev G. I. Stegeman, J. U. Kang, J. S. Aitchison: J.Opt.Soc.Am. B 14, 880 (1997).

    Article  ADS  Google Scholar 

  29. A. V. Buryak and N. N. Akhmediev: Opt. Comm. 110, 287 (1994).

    Article  ADS  Google Scholar 

  30. J. M. Soto-Crespo, N. N. Akhmediev and A. Ankiewicz: J.Opt.Soc.Am. B 12, 1100 (1995).

    Article  ADS  Google Scholar 

  31. N. N. Akhmediev, A. V. Buryak, J. M. Soto-Crespo and D. R. Andersen: J.Opt.Soc.Am. B 12, 434 (1995).

    Article  ADS  Google Scholar 

  32. Y. Chen: Phys. Rev. E 57, 3542 (1998).

    Article  ADS  Google Scholar 

  33. D. Artigas, L. Torner and N. N. Akhmediev: Opt. comm. 143, 322 (1997).

    Article  ADS  Google Scholar 

  34. N. N. Akhmediev, M. P. Das and A. Vagov: Condensed Matter Theories 12, 17 (1997). Ed. J. W. Clark and P. V. Panat

    Google Scholar 

  35. A. V. Buryak and N. N. Akhmediev: Phys. Rev. E 51, 3572 (1995).

    Article  ADS  Google Scholar 

  36. N. N. Akhmediev: Sov. Phys. JETP 56, 299 (1982).

    Google Scholar 

  37. F. V. Kusmartsev: Phys. Rep. 183, 1 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  38. N. Akhmediev, A. Ankiewicz and R. Grimshaw: Phys. Rev. E 59, 6088 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  39. A. A. Kolokolov: Zh. Prikl. Mekh. Tekh. Fiz. 3, 152 (1973). English translation pp.426

    Google Scholar 

  40. M. Grillakis, J. Shatah and W. Strauss: J. Functional Analysis 74, 160 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  41. C. K. R. T. Jones and J. V. Moloney: Phys. Lett. A 117, 175 (1986).

    Article  ADS  Google Scholar 

  42. D. J. Mitchell and A. W. Snyder: J.Opt.Soc.Am. B 10, 1572 (1993).

    Article  ADS  Google Scholar 

  43. I. V. Barashenkov, D. E. Pelinovsky and E. V. Zemlyanaya: Phys. Rev. Lett, 80, 5117 (1998).

    Article  ADS  Google Scholar 

  44. H. T. Tran, J. D. Mitchell, N. N. Akhmediev and A. Ankiewicz: Opt. Comm. 93, 227 (1992).

    Article  ADS  Google Scholar 

  45. I. V. Barashenkov, M. M. Bogdan and T. Zhanlav: in ”Nonlinear World”, Proceedings of the Fourth International Workshop on Nonlinear and Turbulent Processes in Physics, Kiev 1989. Edited by V. G. Bar’yakhtar et al. World Scientific, Singapore, 1990, p.3; I. V. Barashenkov, M. M. Bogdan and V. E. Korobov: Europhysics Letters 15, 113 (1991).

    Google Scholar 

  46. N. N. Akhmediev, A. Ankiewicz and H. T. Tran: J.Opt.Soc.Am. B 10 230 (1993).

    Article  ADS  Google Scholar 

  47. A. D. Boardman and K. Xie: Phys. Rev. E 55, 1899 (1997).

    Article  ADS  Google Scholar 

  48. A. D. Boardman, P. Bontemps and K. Xie: J. Opt. Soc. Am. B 14, 1 (1997).

    Google Scholar 

  49. N. N. Akhmediev and A. V. Buryak: J.Opt.Soc.Am. B 11, 804 (1994).

    Article  ADS  Google Scholar 

  50. L. Torner and G. I. Stegeman: J. Opt. Soc. Am. B (special issue), 14, 3127 (1997).

    Article  ADS  Google Scholar 

  51. L. Torner, D. Mihalache, M. C. Santos and N. N. Akhmediev: J.Opt.Soc.Am. B 15, 1476 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  52. P. Agin and G.I. Stegeman: J. Opt. Soc. Am. B (special issue), 14, 3162 (1997).

    Article  ADS  Google Scholar 

  53. N. N. Akhmediev and J. M. Soto-Crespo: Phys. Rev. A 47, 1358 (1993).

    Article  ADS  Google Scholar 

  54. D. E. Edmundson and R. H. Enns: Opt. Lett. 17, 586 (1992).

    Article  ADS  Google Scholar 

  55. S. Blair and K. Wagner: Optical and Quantum Electronics 30, 697 (1998).

    Article  Google Scholar 

  56. N. Bekki and K. Nozaki: Phys. Lett. A 110, 133 (1985).

    Article  ADS  Google Scholar 

  57. W. Van Saarloos and P. C. Hohenberg: Physica D 56, 303 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  58. N. N. Akhmediev and V. V. Afanasjev: Phys. Rev. Lett. 75, 2320 (1995).

    Article  ADS  Google Scholar 

  59. V. B. Taranenko, K. Staliunas and C. O. Weiss: Phys. Rev. A 56, 1582 (1997).

    Article  ADS  Google Scholar 

  60. W. J. Firth and A. J. Scroggie: Phys. Rev. Lett. 76, 1623 (1996).

    Article  ADS  Google Scholar 

  61. N. A. Kaliteevstii, N. N. Rozanov and S. V. Fedorov: Optics and Spectroscopy 85, 533 (1998).

    Google Scholar 

  62. D. Michaelis, U. Peschel and F. Lederer: Opt. Lett. 23, 1814 (1998).

    Article  ADS  Google Scholar 

  63. A. Hasegawa, and Y. Kodama: Solitons in optical communications, (Oxford University Press, New York, 1995).

    MATH  Google Scholar 

  64. J. D. Moores: Opt. Comm. 96, 65 (1993).

    Article  ADS  Google Scholar 

  65. N. N. Akhmediev, V. V. Afanasjev and J. M. Soto-Crespo: Phys. Rev. E 53, 1190 (1996).

    Article  ADS  Google Scholar 

  66. C. Normand, Y. Pomeau: Rev. Mod. Phys. 49, 581 (1977).

    Article  ADS  MathSciNet  Google Scholar 

  67. P. Kolodner: Phys. Rev. A 44, 6466 (1991).

    Article  ADS  Google Scholar 

  68. R. Graham: Fluctuations, Instabilities and Phase Transitions, (ed. T. Riste, Springer-Verlag, Berlin, 1975).

    Google Scholar 

  69. C. O. Weiss: Phys. Rep. 219, 311 (1992).

    Article  ADS  Google Scholar 

  70. P. K. Jakobsen, J. V. Moloney, A. C. Newell and R. Indik: Phys. Rev. A 45, 8129 (1992).

    Article  ADS  Google Scholar 

  71. G. K. Harkness, W. J. Firth, J. B. Geddes, J. V. Moloney and E. M. Wright: Phys. Rev. A, 50, 4310 (1994).

    Article  ADS  Google Scholar 

  72. G.-L. Oppo, G. D’Alessandro and W. J. Firth: Phys. Rev. A 44, 4712 (1991).

    Article  ADS  Google Scholar 

  73. P.-S. Jian, W. E. Torruellas, M. Haelterman, S. Trillo, U. Peschel and F. Lederer: Opt. Lett. 24, 400 (1999).

    Article  ADS  Google Scholar 

  74. A. M. Dunlop, E. M. Wright and W. J. Firth: Opt. Comm. 147, 393 (1998).

    Article  ADS  Google Scholar 

  75. J. Satsuma and N. Yajima: Progr. Theor. Phys. Suppl. 55, 284 (1974).

    Article  ADS  MathSciNet  Google Scholar 

  76. V. S. Grigoryan and T. C. Muradyan: J.Opt.Soc.Am. B 8, 1757 (1991).

    Article  ADS  Google Scholar 

  77. N. N. Akhmediev, M. J. Lederer and B. Luther-Davies: Phys. Rev. E 57, 3664 (1998).

    Article  ADS  Google Scholar 

  78. N. R. Pereira and L. Stenflo: Phys. Fluids 20, 1733 (1977).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  79. K. Nozaki and N. Bekki: Phys. Soc. Japan. 53, 1581 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  80. P.-A. Bélanger, L. Gagnon and C. Paré: Opt. Lett. 14, 943 (1989).

    Article  ADS  Google Scholar 

  81. J. M. Soto-Crespo, N. Akhmediev and A. Ankiewicz: Phys. Rev. Lett. 85, 2937 (2000).

    Article  ADS  Google Scholar 

  82. R. Conte and M. Musette: Physica D 69, 1 (1993).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  83. B. A. Malomed: Physica D 29, 155 (1987).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  84. O. Thual and S. Fauve: J. Phys. 49, 1829 (1988).

    Article  Google Scholar 

  85. H. R. Brand and R. J. Deissler: Phys. Rev. Lett. 63, 2801 (1989).

    Article  ADS  Google Scholar 

  86. V. Hakim, P. Jakobsen and Y. Pomeau: Europhys. Lett. 11, 19 (1990).

    Article  ADS  Google Scholar 

  87. W. Van Saarloos and P. C. Hohenberg: Phys. Rev. Lett. 64, 749 (1990).

    Article  ADS  Google Scholar 

  88. P. Marcq, H. Chaté and R. Conte: Physica D 73, 305 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Akhmediev, N.N., Ankiewicz, A. (2002). Solitons Around Us: Integrable, Hamiltonian and Dissipative Systems. In: Porsezian, K., Kuriakose, V.C. (eds) Optical Solitons. Lecture Notes in Physics, vol 613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36141-3_5

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  • DOI: https://doi.org/10.1007/3-540-36141-3_5

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