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Journal of Experimental and Theoretical Physics

, Volume 127, Issue 5, pp 903–911 | Cite as

Wave Breaking in Dispersive Fluid Dynamics of the Bose–Einstein Condensate

  • A. M. KamchatnovEmail author
Article

Abstract

The problem of wave breaking during its propagation in the Bose–Einstein condensate to a stationary medium is considered for the case when the initial profile at the breaking instant can be approximated by a power function of the form (–x)1/n. The evolution of the wave is described by the Gross–Pitaevskii equation so that a dispersive shock wave is formed as a result of breaking; this wave can be represented using the Gurevich–Pitaevskii approach as a modulated periodic solution to the Gross–Pitaevskii equation, and the evolution of the modulation parameters is described by the Whitham equations obtained by averaging the conservation laws over fast oscillations in the wave. The solution to the Whitham modulation equations is obtained in closed form for n = 2, 3, and the velocities of the dispersion shock wave edges for asymptotically long evolution times are determined for arbitrary integers n > 1. The problem considered here can be applied for describing the generation of dispersion shock waves observed in experiments with the Bose–Einstein condensate.

Notes

REFERENCES

  1. 1.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Fizmatlit, Moscow, 2006; Pergamon, New York, 1987).Google Scholar
  2. 2.
    R. Courant and K. Friedrichs, Supersonic Flow and Shock Waves (Interscience, New York, 1948).zbMATHGoogle Scholar
  3. 3.
    T. B. Benjamin and M. J. Lighthill, Proc. R. Soc. London, Ser. A 224, 448 (1954).ADSCrossRefGoogle Scholar
  4. 4.
    R. Z. Sagdeev, in Problems of Plasma Theory, Ed. by M. A. Leontovich (Moscow, 1964), No. 4, p. 20 [in Russian].Google Scholar
  5. 5.
    A. V. Gurevich and L. P. Pitaevskii, Sov. Phys. JETP 38, 291 (1973).ADSGoogle Scholar
  6. 6.
    G. B. Whitham, Proc. R. Soc. London, Ser. A 283, 238 (1965).ADSCrossRefGoogle Scholar
  7. 7.
    G. V. Potemin, Usp. Mat. Nauk 43, 39 (1988).MathSciNetGoogle Scholar
  8. 8.
    A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations. An Introductory Course (World Scientific, Singapore, 2000).CrossRefzbMATHGoogle Scholar
  9. 9.
    G. A. El and M. A. Hoefer, Phys. D (Amsterdam, Neth.) 333, 11 (2016).Google Scholar
  10. 10.
    E. P. Gross, Nuovo Cimento 20, 454 (1961).CrossRefGoogle Scholar
  11. 11.
    L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961).MathSciNetGoogle Scholar
  12. 12.
    T. Tsuzuki, J. Low Temp. Phys. 4, 441 (1971).ADSCrossRefGoogle Scholar
  13. 13.
    A. V. Gurevich and A. L. Krylov, Sov. Phys. JETP 65, 944 (1987).Google Scholar
  14. 14.
    V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 37, 823 (1973).ADSGoogle Scholar
  15. 15.
    M. G. Forest and J. E. Lee, in Oscillation Theory, Computation, and Methods of Compensated Compactness, Ed. by C. Dafermos et al., Vol. 2 of IMA Volumes on Mathematics and Its Applications (Springer, New York, 1986).Google Scholar
  16. 16.
    M. V. Pavlov, Teor. Mat. Fiz. 71, 351 (1987).CrossRefGoogle Scholar
  17. 17.
    G. A. El, V. V. Geogjaev, A. V. Gurevich, and A. L. Krylov, Phys. D (Amsterdam, Neth.) 87, 186 (1995).Google Scholar
  18. 18.
    A. M. Kamchatnov, R. A. Kraenkel, B. A. Umarov, Phys. Rev. E 66, 036609 (2002).ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    E. A. Cornell, Talk at NATO Advanced Workshop on Nonlinear Waves: Classical and Quantum Aspects, Lisbon, 2003.Google Scholar
  20. 20.
    M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, Phys. Rev. A 74, 023623 (2006).ADSCrossRefGoogle Scholar
  21. 21.
    A. M. Kamchatnov, A. Gammal, and R. A. Kraenkel, Phys. Rev. A 69, 063605 (2004).ADSCrossRefGoogle Scholar
  22. 22.
    M. A. Hoefer, M. J. Ablowitz, and P. Engels, Phys. Rev. Lett. 100, 084504 (2008).ADSCrossRefGoogle Scholar
  23. 23.
    A. M. Kamchatnov and S. V. Korneev, J. Exp. Theor. Phys. 110, 170 (2010).ADSCrossRefGoogle Scholar
  24. 24.
    S. P. Tsarev, Izv. Akad. Nauk SSSR, Ser. Mat. 54, 1048 (1990).Google Scholar
  25. 25.
    A. V. Gurevich, A. L. Krylov, and G. A. El’, Sov. Phys. JETP 74, 957 (1992).Google Scholar
  26. 26.
    O. Wright, Commun. Pure Appl. Math. 46, 421 (1993).CrossRefGoogle Scholar
  27. 27.
    F. R. Tian, Commun. Pure Appl. Math. 46, 1093 (1993).CrossRefGoogle Scholar
  28. 28.
    A. V. Gurevich, A. L. Krylov, and N. G. Mazur, Sov. Phys. JETP 68, 966 (1989).Google Scholar
  29. 29.
    E. T. Whittaker and D. N. Watson, A Course of Modern Analysis (Cambridge Univ., Cambridge, 1927; Fizmatgiz, Moscow, 1963), Vol. 2.Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2018

Authors and Affiliations

  1. 1.Institute of Spectroscopy, Russian Academy of SciencesTroitsk, MoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia

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