Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 961-990
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.
DOI : https://doi.org/10.1051/m2an:2007004
Classification:  35Q60,  65T50,  65M06
Keywords: Landau damping, Zakharov system
@article{M2AN_2006__40_6_961_0,
     author = {Belaouar, R. and Colin, T. and Gallice, G. and Galusinski, C.},
     title = {Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {6},
     year = {2006},
     pages = {961-990},
     doi = {10.1051/m2an:2007004},
     zbl = {1112.76090},
     mrnumber = {2297101},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_6_961_0}
}
Belaouar, R.; Colin, T.; Gallice, G.; Galusinski, C. Theoretical and numerical study of a quasi-linear Zakharov system describing Landau damping. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, pp. 961-990. doi : 10.1051/m2an:2007004. http://www.numdam.org/item/M2AN_2006__40_6_961_0/

[1] H. Added and S. Added, Equation of Langmuir turbulence and nonlinear Schrödinger equation: smoothness and approximation. J. Funct. Anal. 79 (1988) 183-210. | Zbl 0655.76044

[2] B. Bidégaray, On a nonlocal Zakharov equation. Nonlinear Anal. 25 (1995) 247-278. | Zbl 0830.35123

[3] M. Colin and T. Colin, On a quasilinear Zakharov System describing laser-plasma interactions. Diff. Int. Eqs. 17 (2004) 297-330. | Zbl pre05138408

[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). | MR 2263207 | Zbl 1133.35303

[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. | Zbl 0894.35108

[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. | Zbl 0808.35137

[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. | Zbl 0808.35138

[9] R.T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension. Math. Comp. 58 (1992) 83-102. | Zbl 0746.65066

[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. | Zbl 0928.35158

[11] F. Linares, G. Ponce and J.C. Saut, On a degenerate Zakharov system. Bull. Braz. Math. Soc. New Series 36 (2005) 1-23. | Zbl 1070.35087

[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. | Zbl 0842.35116

[13] G.L. Payne, D.R. Nicholson and R.M. Downie, Numerical Solution of the Zakharov Equations. J. Compt. Phys. 50 (1983) 482-498. | Zbl 0518.76122

[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. Plasmas 6 (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. | Zbl 0639.76054

[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-B 289 (1979) 173-176. | Zbl 0431.35077

[19] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse. Appl. Math. Sci. 139, Springer (1999). | MR 1696311 | Zbl 0928.35157

[20] B. Texier, Derivation of the Zakharov equations. Arch. Rat. Mech. Anal. (to appear). | MR 2289864 | Zbl pre05146096

[21] V.E. Zakharov, S.L. Musher and A.M. Rubenchik, Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Reports 129 (1985) 285-366.