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Mathematical models for laser-plasma interaction

Rémi Sentis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...

On caustics associated with Rossby waves

Arthur D. Gorman (1996)

Applications of Mathematics

Rossby wave equations characterize a class of wave phenomena occurring in geophysical fluid dynamics. One technique useful in the analysis of these waves is the geometrical optics, or multi-dimensional WKB technique. Near caustics, e.g., in critical regions, this technique does not apply. A related technique that does apply near caustics is the Lagrange Manifold Formalism. Here we apply the Lagrange Manifold Formalism to study Rossby waves near caustics.

On caustics associated with the linearized vorticity equation

Petya N. Ivanova, Arthur D. Gorman (1998)

Applications of Mathematics

The linearized vorticity equation serves to model a number of wave phenomena in geophysical fluid dynamics. One technique that has been applied to this equation is the geometrical optics, or multi-dimensional WKB technique. Near caustics, this technique does not apply. A related technique that does apply near caustics is the Lagrange Manifold Formalism. Here we apply the Lagrange Manifold Formalism to determine an asymptotic solution of the linearized vorticity equation and to study associated wave...

On the exact WKB analysis of microdifferential operators of WKB type

Takashi Aoki, Takahiro Kawai, Tatsuya Koike, Yoshitsugu Takei (2004)

Annales de l’institut Fourier

We first introduce the notion of microdifferential operators of WKB type and then develop their exact WKB analysis using microlocal analysis; a recursive way of constructing a WKB solution for such an operator is given through the symbol calculus of microdifferential operators, and their local structure near their turning points is discussed by a Weierstrass-type division theorem for such operators. A detailed study of the Berk-Book equation is given in Appendix.

Perturbation singulière en dimension trois : canards en un point pseudo-singulier nœud

Éric Benoît (2001)

Bulletin de la Société Mathématique de France

On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du type { x ˙ = f ( x , y , z , ε ) , y ˙ = g ( x , y , z , ε ) , ε z ˙ = h ( x , y , z , ε ) , où f , g , h sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente h = 0 est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants...

Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites

Olivier Guès (1995)

Annales de l'institut Fourier

Ce travail concerne le problème de Cauchy-Dirichlet pour des systèmes hyperboliques semilinéaires multidimensionnels perturbés par une “petite viscosité". Les solutions considérées sont C ∞ et locales en temps, le but étant de décrire le comportement de la solution lorsque le paramètre de viscosité ( ϵ > 0 ) tend vers zéro. Il s’agit d’un problème de perturbation singulière pour lequel une “couche limite" se forme au voisinage du bord. Par des méthodes inspirées de l’optique géométrique non linéaire, nous...

Renormalization of exponential sums and matrix cocycles

Alexander Fedotov, Frédéric Klopp (2004/2005)

Séminaire Équations aux dérivées partielles

In this paper, we present a new point of view on the renormalization of some exponential sums stemming from number theory. We generalize this renormalization procedure to study some matrix cocycles arising in spectral problems of quantum mechanics

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