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A singular perturbation problem in a systemof nonlinear Schrödinger equation occurring in Langmuir turbulence

Published online by Cambridge University Press:  15 April 2002

Cédric Galusinski
Cédric Galusinski*
Affiliation:
Université Bordeaux I, Mathématiques Appliquées Bordeaux, ESA 5466 CNRS, 351 cours de la libération, 33400 Talence, France. (galusins@math.u-bordeaux.fr)
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Abstract

The aim of this work is to establish, from amathematical point of view, the limit α → +∞ in the system $i \partial_t E+\nabla (\nabla . E)-\alpha^2 \nabla \times\nabla \times E =-|E|^{2\sigma}E,$ where $E:{\ensuremath{{\Bbb R}}}^3\rightarrow{\mathbb C}^3$ . This corresponds to an approximationwhich is made in the context of Langmuir turbulence in plasmaPhysics. The L 2-subcritical σ (that is σ ≤ 2/3)and the H 1-subcritical σ (that is σ ≤ 2) arestudied. In the physical case σ = 1, the limit is then studied for the $H^1({\ensuremath{{\Bbb R}}}^3)$ norm.

Keywords

Nonlinear Schrödinger equationsingularperturbation.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 109 - 125
Copyright
© EDP Sciences, SMAI, 2000

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References

Bergé, L. and Colin, T., A singular perturbation problem for an envelope equation in plasma physics. Physica D 84 (1995) 437-459. CrossRef
Colin, T., On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occurring in plasma Physics. Differential and Integral Equations 6 (1993) 1431-1450.
R.O. Dendy, Plasma dynamics. Oxford University Press, New York (1990).
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Parts I, II. J. Funct. Anal. 32 (1979) 1-32, 33-71; Part III Ann. Inst. H. Poincaré A 28 (1978) 287-316.
Ginibre, J. and Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal Non Linéaire 2 (1985) 309-402. CrossRef
E.M. Stein, Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton, New Jersey (1970).