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

R. Belaouar; T. Colin; G. Gallice; C. Galusinski

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 40, Issue: 6, page 961-990
  • ISSN: 0764-583X

Abstract

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In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the 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.

How to cite

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Belaouar, R., et al. "Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping." ESAIM: Mathematical Modelling and Numerical Analysis 40.6 (2007): 961-990. <http://eudml.org/doc/194347>.

@article{Belaouar2007,
abstract = { In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the 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. },
author = {Belaouar, R., Colin, T., Gallice, G., Galusinski, C.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Landau damping; Zakharov system.; electron diffusion equation; existence theorems; finite difference scheme},
language = {eng},
month = {2},
number = {6},
pages = {961-990},
publisher = {EDP Sciences},
title = {Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping},
url = {http://eudml.org/doc/194347},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Belaouar, R.
AU - Colin, T.
AU - Gallice, G.
AU - Galusinski, C.
TI - Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/2//
PB - EDP Sciences
VL - 40
IS - 6
SP - 961
EP - 990
AB - In this paper, we study a Zakharov system coupled to an electron diffusion equation in order to describe laser-plasma interactions. Starting from the 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.
LA - eng
KW - Landau damping; Zakharov system.; electron diffusion equation; existence theorems; finite difference scheme
UR - http://eudml.org/doc/194347
ER -

References

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  1. H. Added and S. Added, Equation of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation. J. Funct. Anal.79 (1988) 183–210.  Zbl0655.76044
  2. B. Bidégaray, On a nonlocal Zakharov equation. Nonlinear Anal.25 (1995) 247–278.  Zbl0830.35123
  3. M. Colin and T. Colin, On a quasilinear Zakharov System describing laser-plasma interactions. Diff. Int. Eqs.17 (2004) 297–330.  Zbl1174.35528
  4. 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).  Zbl1133.35303
  5. J.-L. Delcroix and A. Bers, Physique des plasmas 1, 2. Inter Editions-Editions du CNRS (1994).  
  6. J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system. J. Funct. Anal.151 (1997) 384–436.  Zbl0894.35108
  7. L. Glangetas and F. Merle, Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys.160 (1994) 173–215.  Zbl0808.35137
  8. L. Glangetas and F. Merle, Concentration properties of blow up solutions and instability results for Zakharov equation in dimension two. II. Comm. Math. Phys.160 (1994) 349–389.  Zbl0808.35138
  9. R.T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension. Math. Comp.58 (1992) 83–102.  Zbl0746.65066
  10. C.E. Kenig, G. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math.134 (1998) 489–545.  Zbl0928.35158
  11. F. Linares, G. Ponce and J.C. Saut, On a degenerate Zakharov system. Bull. Braz. Math. Soc. New Series36 (2005) 1–23.  Zbl1070.35087
  12. T. Ozawa and Y. Tsutsumi, Existence and smoothing effect of solution for the Zakharov equations. Publ. Res. Inst. Math. Sci.28 (1992) 329–361.  Zbl0842.35116
  13. G.L. Payne, D.R. Nicholson and R.M. Downie, Numerical Solution of the Zakharov Equations. J. Compt. Phys.50 (1983) 482–498.  Zbl0518.76122
  14. 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.  
  15. D.A. Russel, D.F. Dubois and H.A. Rose. Nonlinear saturation of simulated Raman scattering in laser hot spots. Phys. Plasmas6 (1999) 1294–1317.  
  16. K.Y. Sanbomatsu, Competition between Langmuir wave-wave and wave-particule interactions. Ph.D. thesis, University of Colorado, Department of Astrophysical (1997).  
  17. S. Schochet and M. Weinstein, The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys.106 (1986) 569–580.  Zbl0639.76054
  18. C. Sulem and P.-L. Sulem, Quelques résultats de régularité pour les équations de la turbulence de Langmuir. C. R. Acad. Sci. Paris Sér. A-B289 (1979) 173–176.  Zbl0431.35077
  19. C. Sulem and P.-L. Sulem, The nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Appl. Math. Sci.139, Springer (1999).  Zbl0928.35157
  20. B. Texier, Derivation of the Zakharov equations. Arch. Rat. Mech. Anal. (to appear).  Zbl1166.35379
  21. V.E. Zakharov, S.L. Musher and A.M. Rubenchik, Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Reports129 (1985) 285–366.  

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