Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T10:35:51.222Z Has data issue: false hasContentIssue false
  • English
  • Français

Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping

Published online by Cambridge University Press:  15 February 2007

R. Belaouar ,
T. Colin ,
G. Gallice  and
C. Galusinski
R. Belaouar
Affiliation:
SIS, CEA CESTA, BP 2, 33114 Le Barp, France. Mathématiques Appliquées de Bordeaux, UMR CNRS 5466 et CEA LRC M03, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France. [email protected].
T. Colin
Affiliation:
Mathématiques Appliquées de Bordeaux, UMR CNRS 5466 et CEA LRC M03, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France. [email protected]. INRIA Futurs, project MC2.
G. Gallice
Affiliation:
SIS, CEA CESTA, BP 2, 33114 Le Barp, France.
C. Galusinski
Affiliation:
Mathématiques Appliquées de Bordeaux, UMR CNRS 5466 et CEA LRC M03, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France. [email protected]. INRIA Futurs, project MC2.
Get access

Abstract

In this paper, we study a Zakharov system coupled to an electrondiffusion equation in order to describe laser-plasma interactions. Starting fromthe Vlasov-Maxwell system, we derive a nonlinear Schrödinger like system which takes into account the energy exchanged between the plasma waves and the electrons via Landau damping. Two existence theorems are established in a subsonic regime. Using a time-splitting, spectral discretizations for the Zakharov system and a finite difference scheme for the electron diffusion equation, we perform numerical simulations and show how Landau damping works quantitatively.

Keywords

Landau dampingZakharov system.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 961 - 990
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Added, H. and Added, S., Equation of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation. J. Funct. Anal. 79 (1988) 183–210. CrossRef
Bidégaray, B., On a nonlocal Zakharov equation. Nonlinear Anal. 25 (1995) 247–278. CrossRef
Colin, M. and Colin, T., On a quasilinear Zakharov System describing laser-plasma interactions. Diff. Int. Eqs. 17 (2004) 297–330.
T. Colin and G. Metivier, Instabilities in Zakharov Equations for Laser Propagation in a Plasma, Phase Space Analysis of PDEs, A. Bove, F. Colombini, and D. Del Santo, Eds., Progress in Nonlinear Differential Equations and Their Applications, Birkhauser (2006).
J.-L. Delcroix and A. Bers, Physique des plasmas 1, 2. Inter Editions-Editions du CNRS (1994).
Ginibre, J., Tsutsumi, Y. and Velo, G., On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 (1997) 384–436. CrossRef
Glangetas, L. and Merle, F., Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys. 160 (1994) 173–215. CrossRef
Glangetas, L. and Merle, F., Concentration properties of blow up solutions and instability results for Zakharov equation in dimension two. II. Comm. Math. Phys. 160 (1994) 349–389. CrossRef
Glassey, R.T., Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension. Math. Comp. 58 (1992) 83–102. CrossRef
Kenig, C.E., Ponce, G. and Vega, L., Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math. 134 (1998) 489–545. CrossRef
Linares, F., Ponce, G. and Saut, J.C., On a degenerate Zakharov system. Bull. Braz. Math. Soc. New Series 36 (2005) 1–23. CrossRef
Ozawa, T. and Tsutsumi, Y., Existence and smoothing effect of solution for the Zakharov equations. Publ. Res. Inst. Math. Sci. 28 (1992) 329–361. CrossRef
Payne, G.L., Nicholson, D.R. and Downie, R.M., Numerical Solution of the Zakharov Equations. J. Compt. Phys. 50 (1983) 482–498. CrossRef
G. Riazuelo. Étude théorique et numérique de l'influence du lissage optique sur la filamentation des faisceaux lasers dans les plasmas sous-critiques de fusion inertielle. Ph.D. thesis, University of Paris XI.
Russel, D.A., Dubois, D.F. and Rose, H.A.. Nonlinear saturation of simulated Raman scattering in laser hot spots. Phys. Plasmas 6 (1999) 1294–1317. CrossRef
K.Y. Sanbomatsu, Competition between Langmuir wave-wave and wave-particule interactions. Ph.D. thesis, University of Colorado, Department of Astrophysical (1997).
Schochet, S. and Weinstein, M., The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys. 106 (1986) 569–580. CrossRef
Sulem, C. and Sulem, P.-L., Quelques résultats de régularité pour les équations de la turbulence de Langmuir. C. R. Acad. Sci. Paris Sér. A-B 289 (1979) 173–176.
C. Sulem and P.-L. Sulem, The nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Appl. Math. Sci. 139, Springer (1999).
B. Texier, Derivation of the Zakharov equations. Arch. Rat. Mech. Anal. (to appear).
Zakharov, V.E., Musher, S.L. and Rubenchik, A.M., Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Reports 129 (1985) 285–366. CrossRef