Bifurcations and stability of internal solitary waves

  • D. S. Agafontsev
  • F. Dias
  • E. A. Kuznetsov
Article

Abstract

We study both supercritical and subcritical bifurcations of internal solitary waves propagating along the interface between two deep ideal fluids. We derive a generalized nonlinear Schrödinger equation to describe solitons near the critical density ratio corresponding to transition from subcritical to supercritical bifurcation. This equation takes into account gradient terms for the four-wave interactions (the so-called Lifshitz term and a nonlocal term analogous to that first found by Dysthe for pure gravity waves), as well as the six-wave nonlinear interaction term. Within this model, we find two branches of solitons and analyze their Lyapunov stability.

PACS numbers

05.45.Yv 47.55.-t 47.90.+a 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. S. Longuet-Higgins, J. Fluid Mech. 200, 451 (1989).CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    G. Iooss and K. Kirchgässner, C. R. Acad. Sci. 311, 265 (1991).Google Scholar
  3. 3.
    J.-M. Vanden-Broeck and F. Dias, J. Fluid Mech. 240, 549 (1992).CrossRefADSMathSciNetMATHGoogle Scholar
  4. 4.
    F. Dias and G. Iooss, Physica D (Amsterdam) 65, 399 (1993).ADSMathSciNetMATHGoogle Scholar
  5. 5.
    M. S. Longuet-Higgins, J. Fluid Mech. 252, 703 (1993).CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    T. R. Akylas, Phys. Fluids 5, 789 (1993).CrossRefADSMATHGoogle Scholar
  7. 7.
    F. Dias and E. A. Kuznetsov, Phys. Lett. A 263, 98 (1999).CrossRefADSMathSciNetMATHGoogle Scholar
  8. 8.
    E. Falcon, C. Laroche, and S. Fauve, Phys. Rev. Lett. 89, 204501 (2002).Google Scholar
  9. 9.
    F. Dias and G. Iooss, Eur. J. Mech. B/Fluids 15, 367 (1996).MathSciNetMATHGoogle Scholar
  10. 10.
    L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics, 4th ed. (Nauka, Moscow, 1995; Butterworth, London, 1999), Part 1, p. 521.Google Scholar
  11. 11.
    E. A. Kuznetsov, Zh. Éksp. Teor. Fiz. 116, 299 (1999) [JETP 89, 163 (1999)].Google Scholar
  12. 12.
    K. B. Dysthe, Proc. R. Soc. London, Ser. A 369, 105 (1979).ADSMATHGoogle Scholar
  13. 13.
    V. E. Zakharov, Prikl. Mekh. Tekh. Fiz. 9(2), 86 (1968) [J. Appl. Mech. Tech. Phys. 9, 190 (1968)].Google Scholar
  14. 14.
    V. E. Zakharov and E. A. Kuznetsov, Usp. Fiz. Nauk 167, 1137 (1997) [Phys. Usp. 40, 1087 (1997)].CrossRefGoogle Scholar
  15. 15.
    V. E. Zakharov and E. A. Kuznetsov, Zh. Éksp. Teor. Fiz. 113, 1892 (1998) [JETP 86, 1035 (1998)].Google Scholar
  16. 16.
    V. M. Kontorovich, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19, 872 (1976); E. A. Kuznetsov and M. D. Spector, Zh. Éksp. Teor. Fiz. 71, 262 (1976) [Sov. Phys. JETP 44, 136 (1976)].Google Scholar
  17. 17.
    E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, Phys. Rep. 142, 103 (1986).CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    D. S. Agafontsev, F. Dias, and E. A. Kuznetsov, physics/0512006.Google Scholar
  19. 19.
    V. I. Petviashvili, Fiz. Plazmy (Moscow) 2, 469 (1976) [Sov. J. Plasma Phys. 2, 247 (1976)].Google Scholar
  20. 20.
    I. G. Vakhitov and A. A. Kolokolov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, 1020 (1973) [Radiophys. Quantum Electron. 16, 783 (1973)].Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • D. S. Agafontsev
    • 1
  • F. Dias
    • 2
  • E. A. Kuznetsov
    • 1
  1. 1.Landau Institute of Theoretical PhysicsMoscowRussia
  2. 2.Centre de Mathématiques et de Leurs ApplicationsEcole Normale Supérieure de CachanCachan cedexFrance

Personalised recommendations