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N-soliton train and generalized complex Toda chain for the Manakov system

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Abstract

We analyze the dynamical behavior of the N-soliton train of the Manakov system and of the vector NLS equation in the adiabatic approximation. We prove that the dynamics of the N-soliton train in both cases are described by a generalized version of the complex Toda chain model. This fact can be used to predict the asymptotic regimes of the N-soliton train provided the initial soliton parameters are given.

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References

  1. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Acad. Press, San Diego, Calif. (2003).

    Google Scholar 

  2. C. Desem and P. L. Chu, “Soliton-soliton interactions,” in: Optical Solitons: Theory and Experiment (J. R. Taylor, ed.), Cambridge Univ. Press, Cambridge (1992), p. 127; I. M. Uzunov, V. D. Stoev, and T. I. Tzoleva, Opt. Lett., 17, 1417 (1992).

    Google Scholar 

  3. M. Suzuki, H. Toga, N. Edagawa, H. Tanaka, S. Yamamote, and S. Akiba, Electron. Lett., 30, 1083 (1994).

    Article  Google Scholar 

  4. V. I. Karpman and V. V. Solov’ev, Phys. D, 3, 487 (1981).

    Article  MathSciNet  Google Scholar 

  5. I. M. Uzunov, V. S. Gerdjikov, M. Gölles, and F. Lederer, Opt. Commun., 125, 237 (1996).

    Article  Google Scholar 

  6. V. S. Gerdjikov, D. J. Kaup, I. M. Uzunov, and E. G. Evstatiev, Phys. Rev. Lett., 77, 3943 (1996); V. S. Gerdjikov, I. M. Uzunov, E. G. Evstatiev, and G. L. Diankov, Phys. Rev. E, 55, 6039 (1997); V. S. Gerdjikov and I. M. Uzunov, Phys. D, 152–153, 355 (2001).

    Article  ADS  Google Scholar 

  7. V. S. Gerdjikov, E. V. Doktorov, and J. Yang, Phys. Rev. E, 64, 056617 (2001).

    Google Scholar 

  8. V. S. Shchesnovich, Phys. Rev. E, 65, 046614 (2002).

    Google Scholar 

  9. E. V. Doktorov, N. P. Matsuka, and V. M. Rothos, Phys. Rev. E, 69, 056607 (2004).

    Google Scholar 

  10. J. M. Arnold, J. Opt. Soc. Amer. A, 15, 1450 (1998); Phys. Rev. E, 60, 979 (1999).

    ADS  MathSciNet  Google Scholar 

  11. S. V. Manakov, Sov. Phys. JETP, 38, 693 (1974).

    ADS  MathSciNet  Google Scholar 

  12. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method, Nauka, Moscow (1980); English transl.: S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Plenum, New York (1984).

    MATH  Google Scholar 

  13. M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and Continuous Nonlinear Schrodinger Systems (London Math. Soc. Lect. Notes Ser., Vol. 302), Cambridge Univ. Press, Cambridge (2004).

    Google Scholar 

  14. J. Yang, Phys. Rev. E, 65, 036606 (2002).

    Google Scholar 

  15. M. Midrio, S. Wabnitz, and P. Franco, Phys. Rev. E, 54, 5743 (1996); V. S. Shchesnovich and E. V. Doktorov, Phys. Rev. E, 55, 7626 (1997); T. I. Lakoba and D. J. Kaup, Phys. Rev. E, 56, 6147 (1997); S. M. Baker, J. N. Elgin, and J. Gibbons, Phys. Rev. E, 62, 4325 (1999).

    Article  ADS  Google Scholar 

  16. M. Toda, Theory of Nonlinear Lattices, Springer, Berlin (1989); J. Moser, “Finitely many mass points on the line under the influence of an exponential potential: An integrable system,” in: Dynamical Systems, Theory, and Applications (Lect. Notes Phys., Vol. 38, J. Moser, ed.), Springer, Berlin (1975), p. 467; Adv. Math., 16, 197 (1975).

    MATH  Google Scholar 

  17. V. S. Gerdjikov, E. G. Evstatiev, and R. I. Ivanov, J. Phys. A, 31, 8221 (1998); 33, 975 (2000); arXiv:solvint/9909020v1 (1999).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. D. J. Kaup, V. S. Gerdjikov, E. G. Evstatiev, G. L. Diankov, and I. M. Uzunov, “Criterion and regions of stability for quasi-equidistant soliton trains,” Preprint INRNE-TH-97-4, Inst. Nucl. Res. Nucl. Energy, Sofia (1997); arXiv:solv-int/9708004v1 (1997); V. S. Gerdjikov, E. G. Evstatiev, D. J. Kaup, G. L. Diankov, and I. M. Uzunov, Phys. Lett. A, 241, 323 (1998).

    Google Scholar 

  19. D. Anderson, Phys. Rev. A, 27, 3135 (1983); D. Anderson, M. Lisak, and T. Reichel, Phys. Rev. A, 38, 1618 (1988); B. A. Malomed, Progr. Opt., 43, 69 (2002).

    Article  ADS  Google Scholar 

  20. V. S. Gerdjikov, “N-soliton interactions, the complex Toda chain, and stability of NLS soliton trains,” in: Proc. 16th Intl. Symp. on Electromagnetic Theory (Aristotle Univ. of Thessaloniki, Greece, 1998, E. Kriezis, ed.), Vol. 1 (1998), p. 307.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Nuclear Research and Nuclear Energy, 1784, Sofia, Bulgaria

    V. S. Gerdjikov

  2. Stepanov Institute of Physics, 220072, Minsk, Belarus

    E. V. Doktorov

  3. Institute of Mathematics, National Academy of Sciences of Belarus, 220072, Minsk, Belarus

    N. P. Matsuka

Authors
  1. V. S. Gerdjikov
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  2. E. V. Doktorov
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  3. N. P. Matsuka
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 391–404, June, 2007.

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Gerdjikov, V.S., Doktorov, E.V. & Matsuka, N.P. N-soliton train and generalized complex Toda chain for the Manakov system. Theor Math Phys 151, 762–773 (2007). https://doi.org/10.1007/s11232-007-0062-8

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  • DOI: https://doi.org/10.1007/s11232-007-0062-8

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