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

Cédric Galusinski

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 1, page 109-125
  • ISSN: 0764-583X

Abstract

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The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system i t E + ( . E ) - α 2 × × E = - | E | 2 σ E , where E : 3 3 . This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the H 1 ( 3 ) norm.

How to cite

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Galusinski, Cédric. "A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence." ESAIM: Mathematical Modelling and Numerical Analysis 34.1 (2010): 109-125. <http://eudml.org/doc/197473>.

@article{Galusinski2010,
abstract = { The aim of this work is to establish, from a mathematical 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 approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the $H^1(\{\ensuremath\{\{\Bbb R\}\}\}^3)$ norm. },
author = {Galusinski, Cédric},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear Schrödinger equation; singular perturbation.; existence of limit; Schrödinger system; Langmuir turbulence; plasma},
language = {eng},
month = {3},
number = {1},
pages = {109-125},
publisher = {EDP Sciences},
title = {A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence},
url = {http://eudml.org/doc/197473},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Galusinski, Cédric
TI - A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 1
SP - 109
EP - 125
AB - The aim of this work is to establish, from a mathematical 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 approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the $H^1({\ensuremath{{\Bbb R}}}^3)$ norm.
LA - eng
KW - Nonlinear Schrödinger equation; singular perturbation.; existence of limit; Schrödinger system; Langmuir turbulence; plasma
UR - http://eudml.org/doc/197473
ER -

References

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  1. L. Bergé and T. Colin, A singular perturbation problem for an envelope equation in plasma physics. Physica D84 (1995) 437-459.  Zbl1194.82092
  2. T. Colin, On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occurring in plasma Physics. Differential and Integral Equations6 (1993) 1431-1450.  Zbl0780.35104
  3. R.O. Dendy, Plasma dynamics. Oxford University Press, New York (1990).  
  4. 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 A28 (1978) 287-316.  Zbl0396.35028
  5. J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal Non Linéaire2 (1985) 309-402.  Zbl0586.35042
  6. E.M. Stein, Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton, New Jersey (1970).  Zbl0207.13501

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