Nonlinear models for laser-plasma interaction

Thierry Colin; Mathieu Colin; Guy Métivier

Séminaire Équations aux dérivées partielles (2006-2007)

  • page 1-10

Abstract

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In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.

How to cite

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Colin, Thierry, Colin, Mathieu, and Métivier, Guy. "Nonlinear models for laser-plasma interaction." Séminaire Équations aux dérivées partielles (2006-2007): 1-10. <http://eudml.org/doc/11145>.

@article{Colin2006-2007,
abstract = {In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.},
author = {Colin, Thierry, Colin, Mathieu, Métivier, Guy},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {non-fully dispersive Zakharov system; periodic case},
language = {eng},
pages = {1-10},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Nonlinear models for laser-plasma interaction},
url = {http://eudml.org/doc/11145},
year = {2006-2007},
}

TY - JOUR
AU - Colin, Thierry
AU - Colin, Mathieu
AU - Métivier, Guy
TI - Nonlinear models for laser-plasma interaction
JO - Séminaire Équations aux dérivées partielles
PY - 2006-2007
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
SP - 1
EP - 10
AB - In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.
LA - eng
KW - non-fully dispersive Zakharov system; periodic case
UR - http://eudml.org/doc/11145
ER -

References

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  1. H. Added and S. Added. Equation of Langmuir turbulence and nonlinearSchrödinger equation : smoothness and approximation. J. Funct. Anal., Vol. 79, (1988), 183-210. Zbl0655.76044MR950090
  2. R. Belaouard, T. Colin, G. Gallice, C. Galusinski, Theorical and numerical study of a quasilinear Zakharov system describing Landau damping. M2AN vol. 40, No6, 961-986 (2007). Zbl1112.76090MR2297101
  3. C. Besse. Schéma de relaxation pour l’équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson. C.R. Acad. Sci. Paris. Sér. I Math., Vol. 326, (1998), 1427-1432. Zbl0911.65072
  4. B. Bidégaray. On a nonlocal Zakharov equation. Nonlinear Anal., Vol. 25 (3), (1995), 247-278. Zbl0830.35123MR1336525
  5. M. Colin, T. Colin. On a quasilinear Zakharov System describing laser-plasma interactions. Differential and Integral Equations, 17 (2004), no. 3-4, 297–330. Zbl1174.35528MR2037980
  6. M. Colin and T. Colin, A numerical model for the Raman Amplification for laser-plasma interaction. Journal of Computational and Applied Math. 193 (2006), no. 2, 535–562. Zbl1092.35101MR2229560
  7. M. Colin, T. Colin. Multidimensional Raman instability. Preprint 2007. Zbl1121.74367
  8. T. Colin, On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Differential and Integral Equations, vol 6, Number 6, pp. 1431-1450, November 1993. Zbl0780.35104MR1235204
  9. T. Colin, On the standing waves solutions to a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Physica D, 64, pp. 215-236, 1993. Zbl0780.35105MR1214553
  10. T. Colin, G. Métivier, Instabilities in Zakharov equations for laser propagation in a plasma, in Phase space analysis of PDEs, A. Bove, F. Colombini, D. Del Santo Ed., Progress in Nonlinear Differential equations and their Applications 69, Birkhäuser, 2006. Zbl1133.35303MR2263207
  11. Davey A. and Stewartson K. (1974), On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A 338, pp. 101-10. Zbl0282.76008MR349126
  12. J-L. Delcroix and A. Bers. “Physique des plasmas 1, 2”. Inter Editions-Editions du CNRS, (1994). 
  13. J. Ginibre, Y. Tsutsumi and G. Velo. On the Cauchy problem for the Zakharov system. J. Funct. Anal., Vol. 151, (1997), 384-436. Zbl0894.35108MR1491547
  14. L. Glangetas and F. Merle. Existence of self-similar blow-up solutions for Zakharov equation in dimension two. I. Comm. Math. Phys., Vol. 160 (1), (1994), 173-215. Zbl0808.35137MR1262194
  15. L. Glangetas and F. Merle. Concentration properties of blow up solutions and instability results for Zakharov equation in dimension two. II Comm. Math. Phys., Vol. 160 (2), (1994), 349-389. Zbl0808.35138MR1262202
  16. R.T. Glassey. Convergence of an energy-preserving scheme for the Zakharov equation in one space dimension. Math. of Comput. Vol. 58, Number 197, (1992), 83-102. Zbl0746.65066MR1106968
  17. C.E. Kenig, G. Ponce and L. Vega. Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations. Invent. Math., Vol. 134 (3), (1998), 489-545. Zbl0928.35158MR1660933
  18. P. Linares, G. Ponce, J.-C. Saut, On a degenerate Zakharov system, Bull. Braz. Math. Soc. (N.S.) 36, no. 1, 1–23, (2005). Zbl1070.35087MR2132828
  19. G. Métivier, Space Propagation of Instabilities in Zakharov Equations, preprint 2007. Zbl1143.78355
  20. T. Ozawa and Y. Tsutsumi. Existence and smoothing effect of solution for the Zakharov equations. Publ. Res. Inst. Math. Sci, Vol. 28 (3), (1992), 329-361. Zbl0842.35116MR1184829
  21. G. Riazuelo. Etude théorique et numérique de l’influence du lissage optique sur la filamentation des faisceaux lasers dans les plasmas sous-critiques de fusion inertielle. Thèse de l’Université Paris XI. 
  22. D.A. Russel, D.F. Dubois and H.A. Rose. Nonlinear saturation of simulated Raman scattering in laser hot spots. Physics of Plasmas, Vol. 6 (4), (1999), 1294-1317. 
  23. S. Schochet and M. Weinstein. The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. Comm. Math. Phys., Vol. 106, (1986), 569-580. Zbl0639.76054MR860310
  24. C. Sulem and P-L. Sulem. “The nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse.” Applied Mathematical Sciences 139, Springer, (1999). Zbl0928.35157
  25. 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, Vol. 289 (3), (1979), 173-176. Zbl0431.35077MR552204
  26. B. Texier. Derivation of the Zakharov equations. Archive for Rational Mechanics and Analysis 184 (2007), no.1, 121-183. Zbl05146096MR2289864
  27. V.E. Zakharov, S.L. Musher and A.M. Rubenchik. Hamiltonian approach to the description of nonlinear plasma phenomena. Phys. Reports, Vol. 129, (1985), 285-366. MR824169

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