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

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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.

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References

  1. B. B. Kadomtsev and M. B. Kadomtsev, Usp. Fiz. Nauk, 167, 649–664 (1997).

    Google Scholar 

  2. L. P. Pitaevskii, Ibid., 168, 641–653 (1998).

    Article  Google Scholar 

  3. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys., 71, 463–542 (1999).

    Article  ADS  Google Scholar 

  4. A. J. Leggett, Ibid., 73, 307–356 (2001).

    Article  ADS  Google Scholar 

  5. A. Hasegawa and F. Tappert, Appl. Phys. Lett., 23, 171–172 (1973).

    Article  ADS  Google Scholar 

  6. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett., 45, 1095–1098 (1980).

    Article  ADS  Google Scholar 

  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. V. E. Zakharov and V. S. Synakh, Zh. Eksper. Teor. Fiz., 68, 940–947 (1975).

    Google Scholar 

  9. O. Bang, J. J. Rasmussen, and P. L. Christiansen, Nonlinearity, 7, 205–218 (1994).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. Ya. V. Kartashov, L-C. Crasovan, D. Michalache, and L. Torner, Phys. Rev. Lett., 89, 273902(1-4) (2002).

  11. A. V. Shapovalov and A. Yu. Trifonov, J. Nonlin. Math. Phys., 6, No. 2, 1–12 (1999).

    MathSciNet  Google Scholar 

  12. V. P. Maslov, The Complex WKB Method for Nonlinear Equations [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  13. V. V. Belov and S. Yu. Dobrokhotov, Teor. Matem. Fiz., 92, No. 2, 215–254 (1988).

    MathSciNet  Google Scholar 

  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).

  15. A. V. Borisov, A. Yu. Trifonov, and A. V. Shapovalov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 7, 70–75 (2005).

  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.

  17. V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Int. J. Math. Math. Sci., 32, No. 6, 325–370 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. V. Belov, A. Yu. Trifonov, and A. V. Shapovalov, Teor. Matem. Fiz., 130, 460–492 (2002).

    MathSciNet  Google Scholar 

  19. A. L. Lisok, A. Yu. Trifonov, and A. V. Sapovalov, J. Phys. A.: Math. Gen., 37, 4535–4556 (2004).

    Article  ADS  MATH  Google Scholar 

  20. V. E. Zakharov and A. B. Shabat, Zh. Eksper. Teor. Fiz., 61, 118–134 (1971).

    Google Scholar 

  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 

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Authors and Affiliations

  1. Tomsk State University, Russia

    A. V. Borisov & A. V. Shapovalov

  2. Tomsk Polytechnic University, Russia

    A. Yu. Trifonov

Authors
  1. A. V. Borisov
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  2. A. Yu. Trifonov
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  3. A. V. Shapovalov
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__________

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp.-, July, 2006.

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Borisov, A.V., Trifonov, A.Y. & Shapovalov, A.V. A semiclassical approximation for the nonstationary two-dimensional nonlinear Schrödinger equation with an external field in polar coordinates. Russ Phys J 49, 734–743 (2006). https://doi.org/10.1007/s11182-006-0169-7

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  • DOI: https://doi.org/10.1007/s11182-006-0169-7

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