# 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

## Access Full Article

top## Abstract

top## How to cite

topBelaouar, 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

top- 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
- B. Bidégaray, On a nonlocal Zakharov equation. Nonlinear Anal.25 (1995) 247–278. Zbl0830.35123
- M. Colin and T. Colin, On a quasilinear Zakharov System describing laser-plasma interactions. Diff. Int. Eqs.17 (2004) 297–330. Zbl1174.35528
- 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
- J.-L. Delcroix and A. Bers, Physique des plasmas 1, 2. Inter Editions-Editions du CNRS (1994).
- J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system. J. Funct. Anal.151 (1997) 384–436. Zbl0894.35108
- 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
- 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
- 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
- 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
- F. Linares, G. Ponce and J.C. Saut, On a degenerate Zakharov system. Bull. Braz. Math. Soc. New Series36 (2005) 1–23. Zbl1070.35087
- 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
- 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
- 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.
- 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.
- K.Y. Sanbomatsu, Competition between Langmuir wave-wave and wave-particule interactions. Ph.D. thesis, University of Colorado, Department of Astrophysical (1997).
- 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
- 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
- C. Sulem and P.-L. Sulem, The nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Appl. Math. Sci.139, Springer (1999). Zbl0928.35157
- B. Texier, Derivation of the Zakharov equations. Arch. Rat. Mech. Anal. (to appear). Zbl1166.35379
- V.E. Zakharov, S.L. Musher and A.M. Rubenchik, Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Reports129 (1985) 285–366.

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.