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Mathematisches Institut Wien, Austria, | Title and abstract |
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CEA Bruyeres, France, | Title and abstract |
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ENS Cachan, Antenne Kerlann, France, | Title and abstract |
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Univ. Bordeaux I, France, | Title and abstract |
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Univ. Orsay, France, | Title and abstract |
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Univ. Metz, France, | Title and abstract |
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Univ. Rennes 1, France, | Title and abstract |
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Univ. Orsay, France, | Title and abstract |
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CMAP, Ecole Polytechnique, France, | Title and abstract |
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Univ. Orsay, France, | Title and abstract |
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Univ. Angers, France, | Title and abstract |
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Univ. Bordeaux I, France, | Title and abstract |
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CMAP, Ecole Polytechnique, France, | Title and abstract |
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Univ. Bayreuth, Germany, | Title and abstract |
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Univ. North Carolina, USA, | Title and abstract |
Title:
To be precised.
Abstract:
Title:
Self-guided propagation of femtosecond light pulses in air and in fused silica
.
Abstract:
We report experimental results showing the generic property for ultra-short
laser pulses to form robust filaments and to propagate over several Rayleigh
lengths in the atmosphere and in dielectric media, as silica glasses.
Numerical simulations solving an extended nonlinear Schroedinger equation
coupled with ionization sources describe the self-focusing, the temporal
dispersion and multi-photon ionization of light pulses. These simulations are
found to be in good agreement with the experimental data. They show that a
quasi-dynamic equilibrium between ionization and self-focusing drives the
filamentation process, while temporal dispersion plays a minor role.
Title:
Focalisation d'impulsions non-lin\'eaires radiales dans $\R^{1+3}$
Abstract:
Nous \'etudions la validit\'e de l'optique g\'eom\'etrique pour des
impulsions ultra-courtes solutions d'\'equations d'ondes en dimension
trois d'espace, qui focalisent en un point. Si l'amplitude des donn\'ees
initiales est suffisamment grande, ces ondes subissent des effets
non-lin\'eaires tr\`es forts. Dans le cas des \'equations dissipatives,
les impulsions courtes sont absorb\'ees. Dans le cas des \'equations
accr\'etives, la famille des impulsions devient non born\'ee. Pour une
amplitude plus faible mais critique, l'optique g\'eom\'etrique lin\'eaire
est valide avant le point focal, et apr\`es. La transition entre ces deux
r\'egimes est d\'ecrite par un op\'erateur de diffusion, qui \'elargit le
support des ondes.
Title:
Computation of the interaction of a laser with a gas
Abstract:
We start with the Maxwell-Bloch equations that describe the interaction of a
laser
with a two level medium. We compute the interaction coefficient with respect to
the density of the medium. Then we restrict ourself to the case of a dilute gas.
We present an asymptotic model
and prove some asymptotic results. Then we present some numerical simulations.
Title:
Rigorous semiclassical results for the magnetic response of an electron
gas.
Abstract:
Consider a noninteracting electrons gas in a confining potential and a
magnetic field in arbitrary dimensions. If this gas is in thermal
equilibrium with a reservoir at temperature T>0, one can study its
orbital magnetic response (omitting the spin). One defines a
conveniently "smeared out" magnetization M which will be analyzed from a
semiclassical point of view, namely when the Planck constant h tends to
zero. Then various regimes of temperature T are studied Where M can be
obtained in the form of suitable h-expansions.In particular when T is
of the order of h, oscillations "a la de Haas-Van Alphen" appear, that
can be linked to the classical periodic orbits of the electronic motion.
Title:
Diffraction by an immersed elastic wedge.
Abstract:
We present a numerical method to compute very accurately the acoustic wave
diffracted by an elastic 2D-wedge immersed in an incompressible fluid, in
the limit of the high frequencies. The equations are the linear elasticity
equations inside the wedge and the scalar wave equation inside the fluid.
They are coupled by the continuity of the pressure and of the normal
velocity along the faces of the wedge. \par The principle of the method is
to write the solution with the help of a {\it spectral function} which is
the 1D-Fourier transform of layer potentials along the faces of the wedge.
This meromorphic function is solution of a singular kernel integral
system. This system is solved very accurately using a splitting of the
spectral fuction into two parts. The first part is an explicit meromorphic
function corresponding to successive reflexions of the incoming wave
against the faces. The second one is holomorphic and is computed by a
Galerkin/collocation method.
Title:
Fronti\`eres ombre-lumi\`ere.
Abstract:
On donne une description asymptotique
\`a trois \'echelles de solutions de syst\`emes hyperboliques
non lin\'eaires \`a coefficients variables : la solution
$u^\eps$ est approch\'ee par $\eps^m \sum \eps
^{n/2} u_n (x,\psi(x)/\eps^{1/2},\phi(x)/\eps)$. La quantit\'e $1/\eps$ est la fr\'equence d'oscillation, le
s profils $u_n$ \'etant p\'eriodiques par rapport \`a la variable $\phi/\eps$. Par contre, ils ont un
comportement de type ''marche'' ($0$ en $-\infty$, limite finie en $+
\infty$) en la variable interm\'ediaire $\psi/\eps^{1/2}$, ce qui traduit une transition dans une couche l
imite de taille $\eps^{1/2}$ autour de la surface $\psi=0$ (la variable d'espace-temps $x$ est ${\cal O}(1
)$).
Chaque profil est d\'etermin\'e par un probl\`eme de Cauchy associ\'e \`a un transport non lin\'eair
e \`a l'\'echelle $x$ (optique g\'eom\'etrique) coupl\'e \`a une \'equation de Schr\"odinger non lin
\'eaire (dispersion dans des directions transverses). Le d\'eveloppement asymptotique (valide lors d
e la perturbation des donn\'ees) est construit \`a tout ordre, sous des hypoth\`eses d'imparit\'e de
s non-lin\'earit\'es, et au premier ordre dans le cas de non-lin\'earit\'es quadratiques, pour des s
yst\`emes conservatifs.
Title:
Une formule de Landau-Zener pour les mesures semi-classiques.
Abstract:
Dans cet expos\'e, on \'etudie l'\'evolution des mesures
semi-classiques d'une famille de solutions
d'un syst\`{e}me pr\'esentant un croisement de modes.
A cet effet, on introduit des mesures
semi-classiques \`{a} deux \'echelles qui d\'ecrivent comment la
transform\'ee de Wigner usuelle
se concentre sur des trajectoires rencontrant un croisement.
Des formules
explicites de type Landau-Zener d\'ecrivent les traces de ces mesures
au niveau du croisement.
Title:
Soliton propagation in random media.
Abstract:
The investigation of the competition between randomness and
nonlinearity for wave propagation phenomena in the one-dimensional case
is of great interest for applications in nonlinear optics and optical
transmission systems.
As it is well known, in one-dimensional linear media with random
inhomogeneities strong localization occurs, which means in particular
that the transmitted intensity decays exponentially as a function
of the size of the medium.
On the other hand, in some homogeneous nonlinear media corresponding
to integrable systems, wavepackets called
solitons can propagate without change of form or diminution of speed.
We shall study the transmission of a soliton through a slab of
nonlinear and random medium.
More exactly we shall first consider the one-dimensional continuum
nonlinear Schrodinger (NLS) equation, and we assume that
inhomogeneities affect the potential, the nonlinear coefficient,
and the dispersion.
We study the influence of the random perturbations on the propagation
properties of the integrable NLS equation.
Several asymptotic behaviors can be exhibited
when the amplitudes of the random fluctuations go to zero
and the size of the slab goes to infinity.
The mass of the transmitted soliton may tend to zero exponentially
(as a function of the size of the slab) or following a power law;
or else the soliton may keep its mass, while its velocity slowly
decays to zero.
Second we shall address other integrable systems,
such as a vector NLS equation (the so-called Manakov system)
and the Korteweg-de Vries (KdV) equation.
We shall see that original behaviors are possible.
In particular the emission of a soliton gas is proved
to be a general feature for the random KdV equation.
Title:
From the quasi-static to the dynamic Maxwell model in
micromagnetism.
Abstract:
A commonly used model for ferromagnetic materials in the quasistatic
regime is the Landau-Lifshitz system coupled with the so-called
quasistatic Maxwell equation. By an appropriate scalling, we justify
this approach and we propose a new asymptotic expansion. This
suggest a new numerical method.
Title:
Nonlinearities in Diffractive Optics with Continuous Spectra.
Abstract:
Continuous oscillating spectra are used to modelize a few physical problems which
cannot
be treated with the usual amplitude equations for wave trains. For instance:
Raman scattering,
short pulses and laser with large spectrum.
We propose here a framework which generalizes the known results of diffractive
optics for wave trains
to those more general oscillations.
We particularly focus on the behaviour of the nonlinearities which can be sorted
out using two kinds of
arguments: one relies on the diffractive time-scale of the study, while the other
roughly says that some
nonlinearities have "zero measure", in a sense to be precised.
Title:
Multiscale expansion plus ...
Abstract:
How a further approximation can be taken into account in a multiscale expansion.
1)
The Maxwell-Landau model with damping.
Different models are obtained, depending on the order
of magnitude of the damping.
2)
Cascaded second order non-linearities.
Using a set of adequate assumptions on the linear susceptibility,
and a small deviation angle of the propagation direction,
we can justify the model, and determine
approximate phase and group velocity matching condition.
3)
Magnetostaticwaves propagation in magnetic thin films.
The coefficients of the NLS model appreciably differs
from the values commonly admitted. Especially,
the dispersion coefficient is not the so-called group velocity dispersion.
Title:
Semi-Classical Calculus and a Domain Decomposition Method for a PDE
with a Small Parameter.
Abstract:
The first domain decomposition algorithm was proposed by H. Schwarz
(1890) for studying the well-posedness of the Laplace equation.
Domain decomposition methods are now used for solving very large
scale problems on parallel computers. The original Schwarz method is
very slow. In order to improve it, it has been proposed to modify it
by changing the interface conditions between the subdomains. The
optimal interface conditions (in terms of iteration counts) are
pseudodifferential and involve the Steklov-Poincare (Dirichlet to
Neumann map, DtN) operator. These conditions are very difficult to
implement and are approximated by partial differential operators. By
taking advantage of the small parameter in the PDE, it is possible to
design "low" frequency approximations and analyze the convergence
rate of the resulting algorithm.
Title:
Validity of models for the 2D-water wave problem
- The Nonlinear Schr\"odinger equation -
Abstract:
We consider the validity of the Nonlinear Schr\"odinger equation
for the water wave problem, i.e. first we prove
the existence of modulating pulse solutions for the water
wave problem. These solutions consist of a pulse like
envelope advancing in the laboratory frame and modulating
an underlying spatially and temporarely oscillating wave train.
These solutions correspond to the 1-soliton
solutions of the NLS-equation.
Secondly, we prove exact estimates between the solutions of the
water wave problem and the approximations obtained via
the NLS equation.
Title:
Singular pseudodifferential operators and oscillatory multidimensional
shocks.
Abstract:
We introduce a class of singular pseudodifferential operators permitting
simultaneous microlocalization in both slow and fast variables for some kinds of
singular quasilinear hyperbolic systems. We apply them to prove the existence of
oscillatory multidimensional shocks on a fixed time interval independent of the
wavelength epsilon as epsilon goes to zero.