Solutions to the cold plasma model at resonances
Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 177-222.

Little is known on the mathematical theory of hybrid and cyclotron solutions of Maxwell’s equations with the cold plasma dielectric tensor. These equations arise in magnetized plasmas to model an electromagnetic wave in a Tokamak. The solutions can behave extremely differently from those in vacuum. This work contributes to the local theory of the hybrid resonance by means of an original representation formula based on special functions, a certain eikonal equation and with a careful treatment of the singularity.

La théorie mathématiques des équations de Maxwell avec le tenseur du plasma froid en présence d’une résonance cyclotron ou hybride est peu développée. Ces équations sont présentes pour modéliser la propagation d’une onde électromagnétique dans un plasma magnétique tel que celui d’un Tokamak et les solutions peuvent être très différentes de la propagation dans le vide. Ce travail contribue principalement à la théorie locale de la résonance hybride avec des formules originales de représentation à partir de fonctions spéciales. Ces formules sont obtenues au moyen d’une équation eikonale et d’un traitement spécifique de la singularité.

Published online:
DOI: 10.5802/jep.41
Classification: 34M03, 78A25, 34C11
Keywords: Cold plasma model, resonant Maxwell’s equations, hybrid resonance, eikonal equation, singular solutions of ODEs
Bruno Després 1; Lise-Marie Imbert-Gérard 2; Olivier Lafitte 3

1 Sorbonne Universités, UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions 75252 Paris Cedex 05, France
2 Courant Institute of Mathematical Sciences, New York University 251 Mercer Street, New York, NY 10012-1185, USA
3 Université Paris 13, Sorbonne Paris Cité, LAGA (UMR 7539) 93430 Villetaneuse, France and DM2S/DIR, CEA Saclay 91191 Gif-sur-Yvette cedex, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Bruno Despr\'es and Lise-Marie Imbert-G\'erard and Olivier Lafitte},
     title = {Solutions to the cold plasma model at~resonances},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {177--222},
     publisher = {\'Ecole polytechnique},
     volume = {4},
     year = {2017},
     doi = {10.5802/jep.41},
     zbl = {1378.78013},
     mrnumber = {3611102},
     language = {en},
     url = {}
AU  - Bruno Després
AU  - Lise-Marie Imbert-Gérard
AU  - Olivier Lafitte
TI  - Solutions to the cold plasma model at resonances
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2017
SP  - 177
EP  - 222
VL  - 4
PB  - École polytechnique
UR  -
DO  - 10.5802/jep.41
LA  - en
ID  - JEP_2017__4__177_0
ER  - 
%0 Journal Article
%A Bruno Després
%A Lise-Marie Imbert-Gérard
%A Olivier Lafitte
%T Solutions to the cold plasma model at resonances
%J Journal de l’École polytechnique — Mathématiques
%D 2017
%P 177-222
%V 4
%I École polytechnique
%R 10.5802/jep.41
%G en
%F JEP_2017__4__177_0
Bruno Després; Lise-Marie Imbert-Gérard; Olivier Lafitte. Solutions to the cold plasma model at resonances. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 177-222. doi : 10.5802/jep.41.

[1] - Handbook of mathematical functions with formulas, graphs, and mathematical tables (M. Abramowitz & I. A. Stegun, eds.), Dover Publications, Inc., New York, 1992 | Zbl

[2] A.-S. Bonnet-Ben Dhia, L. Chesnel & P. Ciarlet - “T-coercivity for scalar interface problems between dielectrics and metamaterials”, ESAIM Math. Model. Numer. Anal. 46 (2012) no. 6, p. 1363-1387 | DOI | Numdam | MR | Zbl

[3] M. Brambilla - “The effects of Coulomb collisions on the propagation of cold-plasma waves”, Phys. Plasmas 2 (1995) no. 4, p. 1094-1099 | DOI

[4] M. Brambilla - Kinetic theory of plasma waves. Homogeneous plasmas, International Series of Monographs on Physics, Clarendon Press, 1998

[5] K. G. Budden - Radio waves in the ionosphere: The mathematical theory of the reflection of radio waves from stratified ionised layers, Cambridge University Press, New York, 1961 | MR | Zbl

[6] M. Campos-Pinto & B. Després - “Constructive formulations of resonant Maxwell’s equations” (2016), preprint hal-01278860 | Zbl

[7] Y. Chen & R. Lipton - “Resonance and double negative behavior in metamaterials”, Arch. Rational Mech. Anal. 209 (2013) no. 3, p. 835-868 | DOI | MR | Zbl

[8] E. A. Coddington & N. Levinson - Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955 | Zbl

[9] B. Després, L.-M. Imbert-Gérard & R. Weder - “Hybrid resonance of Maxwell’s equations in slab geometry”, J. Math. Pures Appl. (9) 101 (2014) no. 5, p. 623-659 | DOI | MR | Zbl

[10] R. J. Dumont, C. K. Phillips & D. N. Smithe - “Effects of non-Maxwellian species on ion cyclotron waves propagation and absorption in magnetically confined plasmas”, Phys. Plasmas 12 (1995) no. 4, 042508 pages | DOI

[11] L.-M. Imbert-Gérard - Analyse mathématique et numérique de problémes d’ondes apparaissant dans les plasmas magnétiques, UPMC, Université Paris VI, 2013, Ph. D. Thesis

[12] , ITER organization web page

[13] O. Lafitte, M. Williams & K. Zumbrun - “High-frequency stability of detonations and turning points at infinity”, SIAM J. Math. Anal. 47 (2015) no. 3, p. 1800-1878 | DOI | MR | Zbl

[14] R. W. McKelvey - “The solutions of second order linear ordinary differential equations about a turning point of order two”, Trans. Amer. Math. Soc. 79 (1955), p. 103-123 | DOI | MR | Zbl

[15] W. Rudin - Real and complex analysis, McGraw-Hill Book Co., Inc., New York, 1987 | Zbl

[16] F. da Silva, M. Campos Pinto, B. Després & S. Heuraux - “Stable coupling of the Yee scheme with a linear current model”, J. Comput. Phys. 295 (2015), p. 24-45 | DOI | Zbl

[17] F. da Silva, S. Heuraux, E. Z. Gusakov & A. Popov - “A numerical study of forward- and backscattering signatures on Doppler-reflectometry signals”, IEEE Trans. Plasma Sci. 38 (2010) no. 9, p. 2144 -2149 | DOI

[18] T. H. Stix - The theory of plasma waves, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962 | MR | Zbl

[19] D. G. Swanson - Plasma waves, Academic Press, Inc., Boston, MA, 1989 | MR

[20] F. G. Tricomi - Integral equations, Pure and Applied Mathematics, vol. V, Interscience Publishers, Inc., New York, London, 1957 | Zbl

[21] R. Weder - “A rigorous analysis of high-order electromagnetic invisibility cloaks”, J. Phys A 41 (2008) no. 6, 065207 pages | DOI | MR | Zbl

[22] R. B. White & F. F. Chen - “Amplification and absorption of electromagnetic waves in overdense plasmas”, Plasma Physics 16 (1974) no. 7, 565 pages, anthologized in Laser Interaction with Matter, Series of Selected Papers in Physics, ed. by C. Yamanaka, Phys. Soc. Japan, 1984 | DOI

[23] L. F. Ziebell & R. S. Schneider - “The effective dielectric tensor for electromagnetic waves in inhomogeneous magnetized plasmas and the proper formulation in the electrostatic limit”, Brazilian J. Phys. 34 (2004), p. 1211-1223 | DOI

Cited by Sources: