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Russian Physics Journal

, Volume 49, Issue 7, pp 734–743 | Cite as

A semiclassical approximation for the nonstationary two-dimensional nonlinear Schrödinger equation with an external field in polar coordinates

  • A. V. Borisov
  • A. Yu. Trifonov
  • A. V. Shapovalov
Article
  • 23 Downloads

Abstract

For the two-dimensional nonlinear Schrödinger equation (NSE) with an external field in polar coordinates, semiclassical solutions asymptotic in small parameter ħ, ħ → 0, have been constructed. The solutions are constructed in the class of functions which are concentrated in a neighborhood of a closed plane curve and have the form of the one-soliton solution of the one-dimensional NSE in the “fast” variable.

Keywords

Soliton External Field Principal Term Asymptotic Parameter Semiclassical Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    B. B. Kadomtsev and M. B. Kadomtsev, Usp. Fiz. Nauk, 167, 649–664 (1997).Google Scholar
  2. 2.
    L. P. Pitaevskii, Ibid., 168, 641–653 (1998).CrossRefGoogle Scholar
  3. 3.
    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys., 71, 463–542 (1999).CrossRefADSGoogle Scholar
  4. 4.
    A. J. Leggett, Ibid., 73, 307–356 (2001).CrossRefADSGoogle Scholar
  5. 5.
    A. Hasegawa and F. Tappert, Appl. Phys. Lett., 23, 171–172 (1973).CrossRefADSGoogle Scholar
  6. 6.
    L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett., 45, 1095–1098 (1980).CrossRefADSGoogle Scholar
  7. 7.
    H. Yuen and B. Lake, Nonlinear Dynamics of Deep Water Gravity Waves, in: Advances in Applied Mechanics, Academic Press, NY (1982), Vol. 22, pp. 67–229.Google Scholar
  8. 8.
    V. E. Zakharov and V. S. Synakh, Zh. Eksper. Teor. Fiz., 68, 940–947 (1975).Google Scholar
  9. 9.
    O. Bang, J. J. Rasmussen, and P. L. Christiansen, Nonlinearity, 7, 205–218 (1994).zbMATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Ya. V. Kartashov, L-C. Crasovan, D. Michalache, and L. Torner, Phys. Rev. Lett., 89, 273902(1-4) (2002).Google Scholar
  11. 11.
    A. V. Shapovalov and A. Yu. Trifonov, J. Nonlin. Math. Phys., 6, No. 2, 1–12 (1999).MathSciNetGoogle Scholar
  12. 12.
    V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977).Google Scholar
  13. 13.
    V. V. Belov and S. Yu. Dobrokhotov, Teor. Matem. Fiz., 92, No. 2, 215–254 (1988).MathSciNetGoogle Scholar
  14. 14.
    A. V. Borisov, A. Yu. Trifonov, and A. V. Shapovalov, Mathematics, Computers, Education [in Russian], ed. by G. Yu. Riznichenko, 2, issue 12, 648–659 (2005).Google Scholar
  15. 15.
    A. V. Borisov, A. Yu. Trifonov, and A. V. Shapovalov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 7, 70–75 (2005).Google Scholar
  16. 16.
    A. V. Borisov, A. V. Shapovalov, and A. Yu. Trifonov, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 1, Paper 019 (2005); http://www.emis.de/journals/SIGMA.Google Scholar
  17. 17.
    V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Int. J. Math. Math. Sci., 32, No. 6, 325–370 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Teor. Matem. Fiz., 130, 460–492 (2002).MathSciNetGoogle Scholar
  19. 19.
    A. L. Lisok, A. Yu. Trifonov, and A. V. Sapovalov, J. Phys. A.: Math. Gen., 37, 4535–4556 (2004).CrossRefADSzbMATHGoogle Scholar
  20. 20.
    V. E. Zakharov and A. B. Shabat, Zh. Eksper. Teor. Fiz., 61, 118–134 (1971).Google Scholar
  21. 21.
    V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Problem Method, Plenum, NY (1984).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. V. Borisov
    • 1
  • A. Yu. Trifonov
    • 2
  • A. V. Shapovalov
    • 1
  1. 1.Tomsk State UniversityRussia
  2. 2.Tomsk Polytechnic UniversityRussia

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