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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topGalusinski, 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

top- L. Bergé and T. Colin, A singular perturbation problem for an envelope equation in plasma physics. Physica D84 (1995) 437-459. Zbl1194.82092
- 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
- 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 A28 (1978) 287-316. Zbl0396.35028
- 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
- E.M. Stein, Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton, New Jersey (1970). Zbl0207.13501

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