Skip to main content
SpringerLink
Log in

On the Boyd-Kadomstev System for a Three-Wave Coupling Problem and its Asymptotic Limit

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the Boyd-Kadomstev system which is in particular a model for the Brillouin backscattering in laser-plasma interaction. It couples the propagation of two laser beams, the incoming and the backscattered waves, with an ion acoustic wave which propagates at a much slower speed. The ratio \({\varepsilon}\) between the plasma sound velocity and the (group) velocity of light is small, with typical value of order 10−3. In this paper, we make a rigorous analysis of the behavior of solutions as \({\varepsilon \to 0}\) . This problem can be cast in the general framework of fast singular limits for hyperbolic systems. The main new point which is addressed in our analysis is that the singular relaxation term present in the equation is a nonlinear first order system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ballereau Ph., Casanova M., Duboc F., Dureau D., Jourdren H., Loiseau P., Metral J., Morice O., Sentis R.: Coupling Hydrodynamics with a Paraxial Solver for Laser Propagation. J. Sci. Comp. 33, 1–24 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berger R.L., Still C.H., Williams E.A., Langdon A.B.: On the dominant subdominant behavior of stimulated Raman and Brillouin scattering. Phys. Plasmas 5, 4337 (1998)

    Article  ADS  Google Scholar 

  3. Boyd J.M., Turner J.G.: Lagrangian studies of plasma wave interaction. J. Physics A: Gen. Phys. 5, 881–896 (1972)

    Article  ADS  Google Scholar 

  4. Colin M., Colin T.: On a quasilinear Zakharov system describing laser-plasma interaction. Diff. Int. Eqs. 17, 297–330 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Colin M., Colin T.: A numerical model for the Raman amplification for laser-plasma interaction. J. Compt. Appl. Maths. 193, 535–562 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Colin, M., Colin, T., Métivier, G.: Nonlinear models for laser-plasma interactions. Séminaire X-EDP, Ecole Polytéchnique, X1–X10 (2007)

  7. Joly J.-L., Métivier G., Rauch J.: Transparent Nonlinear Geometric Optics and Maxwell-Bloch Equations. J. Diff. Eq. 166, 175–250 (2000)

    Article  MATH  Google Scholar 

  8. Joly J.-L., Métivier G., Rauch J.: Coherent and focusing multidimensional nonlinear geometric optics. Ann. Sci. Ecole Norm. Sup. 28, 51–113 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Kadomtsev, B.B.: Plasma Turbulence. London-New York: Academic Press, 1965, translated from Russian version published in 1964

  10. Klainerman S., Majda A.: Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Klein, E.M.: Singular integrals and differentiability properties. Princeton, NJ: Princeton Univ. Press, 1970

  12. Pesme D., Laval G., Pellat M.: Parametric instability. Phys. Rev. Lett. 31, 203 (1973)

    Article  ADS  Google Scholar 

  13. Loiseau P. et al.: Laser Beam smoothing induced by stimulated Brillouin scatter. Phys. Rev. Lett. 97, 205001 (2006)

    Article  ADS  Google Scholar 

  14. Métivier, G.: The Mathematics of Nonlinear Optics. Handbook of Differential Equations: Evolutionary Equations, Vol. 5, Amsterdam: Elsevier, 2009

  15. Métivier G., Schochet S.: The incompressible Limit of the Nonisentropic Euler Equations. Arch. Rat. Mech. Anal. 158, 61–90 (2001)

    Article  MATH  Google Scholar 

  16. Mounaix Ph., Pesme D., Casanova M.: Nonlinear reflectivity of an inhomogeneous plasma. Phys. Rev. E 55, 4653–4664 (1997)

    Article  ADS  Google Scholar 

  17. Novikov, S., Zakharov, V.E., et al.: Theory of solitons. New York: Consultant Bureau, 1984

  18. Pesme, D.: Interaction collisionnelle et collective (Chap 2). In: La fusion par Confinement Inertiel I. Interaction laser-matière. R. Dautray-Watteau, ed., Paris: Eyrolles, 1995

  19. Hüller, S., Masson-Laborde, P.E., et al.: Harmonic Decomposition to describe the nonlinear evolution of stimulated Brillouin Scattering Phys. of Plasma, 13, 022703 (2006). Cf. also D. Teychenné, et al.: Model and conservation laws... CEA Internal rep. R-6195, 2008

  20. Sentis R.: Mathematical Models for Laser-Plasma Interaction. ESAIM-Math. Modelling Num. Analysis 39, 275 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sentis R.: On the Boyd-Kadomtsev system for the three-wave coupling. Note C. R. Acad. Sci., Paris, Ser. I, Mathematique 347, 933–938 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schochet S.: Fast singular limits of hyperbolic PDEs. J. Diff. Eqs. 114(1994), 476–512 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tartar, L.: An introduction to Navier-Stokes equation. Berlin: Springer, 2006

Download references

Author information

Authors and Affiliations

  1. IMB, Université de Bordeaux, 33405, Talence Cedex, France

    Guy Métivier

  2. CEA, DAM, DIF Bruyères, 91297, Arpajon, France

    Rémi Sentis

Authors
  1. Guy Métivier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rémi Sentis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rémi Sentis.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Métivier, G., Sentis, R. On the Boyd-Kadomstev System for a Three-Wave Coupling Problem and its Asymptotic Limit. Commun. Math. Phys. 319, 303–330 (2013). https://doi.org/10.1007/s00220-013-1672-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-013-1672-7

Keywords

Search

Navigation