(I3M, Institut de Mathématique et de
Modélisation de Montpellier,
University of Montpellier, France)
Laurent Di Menza
(LMR, Laboratoire de Mathématiques de Reims, University of Reims,
France)
Title: Revisited transparent
conditions for the Schrödinger equation
Abstract:
The problem of well-adapted boundary conditions is essential
when dealing
with a bounded domain for the
numerical resolution of PDE's set on the
whole space. In the ideal
case, the solution is nothing else but the
restriction of the solution to
the computational domain ; the condition
is said to be transparent.
Unfortunately, these conditions are nonlocal
and not easy to implement. In
this talk, we fist recall a few facts about
the design of transparent
conditions for the wave equation and the
Schrödinger equation. We then
address a new formulation for these
conditions leading to a local
expression with illustrations in the
one-dimensional and
two-dimensional linear cases.
Romain Duboscq
(IMT, Institut de Mathématiques de Toulouse, University of Toulouse,
France)
Title: Numerical methods for
Bose-Einstein condensates.
Abstract:
The aim of this talk is to develop
some robust and accurate numerical methods to compute the stationary
states as well as the dynamic of Bose-Einstein condensates (BEC). Most
particularly, we are interested in deterministic and stochastic
Gross-Pitaevskii equations (or systems of Gross-Pitaevskii equations)
which models BEC for different experimental designs. In order to
compute the stationary states of such systems, we have to ensure that
the numerical methods are fast and, more importantly, robust with
respect to some critical parameters involved in the equation (for
instance the rotational speed). Moreover, we wish to obtain methods
that accurately describe the different physical phenomenons that arise
from the different models. We will give some numerical examples
computed by GPELab which is a freely available Matlab toolbox developed
in collaboration with Xavier Antoine (IECL).
Wolf-Patrick
Düll
(Institut für Analysis, Dynamik und Modellierung, Universität
Stuttgart,
Germany)
Title: Justification
of the Nonlinear Schrödinger equation for the evolution of gravity
driven 2D surface water waves in a canal of finite depth
Abstract: In 1968 V.E. Zakharov
derived the Nonlinear Schrödinger equation as an approximation equation
for the two-dimensional water wave problem in the absence of surface
tension, i.e., for the evolution of gravity driven surface water waves,
in order to describe slow temporal and spatial modulations of a
spatially and temporarily oscillating wave packet. In this talk we give
a rigorous proof that the wave packets in the two-dimensional water
wave problem in a canal of finite depth can be accurately approximated
by solutions of the Nonlinear Schrödinger equation.
Ansgar
Jüngel
(Institute for Analysis and Scientific Computing, Vienna University of
Technology, Germany)
Title:
Modeling and analysis of diffusive
quantum fluid equations
Abstract:
Quantum fluid equations are an
interesting alternative to more sophisticated
but computationally very
expensive quantum transport models. In this talk,
two classes of quantum fluid
models are analyzed: quantum drift-diffusion
and quantum Navier-Stokes
equations. These models can be formally derived
from a collisional Wigner
equation in the diffusion limit. The quantum
Navier-Stokes equations can be
also obtained from a dissipative Euler-Lagrange
equation with a quantum action
functional on the space of probability measures,
which allows one to relate
quantum fluid models with the (one-particle)
Schroedinger equation.
The main mathematical
challenge of the diffusive quantum models is the treatment
of the highly nonlinear
higher-order differential operators. The existence
of global weak solutions is
proved by using new analytical tools like entropy
methods, systematic
integration of parts, and osmotic velocity variable
transformation.
Christian Klein
(Institut de Mathématiques de Bourgogne, University of Dijon, France,
to
be confirmed)
Title: Multidomain
spectral method for Schrödinger equations
Abstract: A
multidomain spectral method with compactified exterior domains combined
with stable second and fourth order time integrators is presented for
Schrödinger equations. The numerical approach allows high precision
numerical studies of solutions on the whole real line. At examples for
the linear and cubic nonlinear Schrödinger equation, this code is
compared to transparent boundary conditions and perfectly matched
layers approaches. The code can deal with asymptotically non vanishing
solutions as the Peregrine breather being discussed as a model for
rogue waves. It is shown that the Peregrine breather can be numerically
propagated with essentially machine precision, and that localized
perturbations of this solution can be studied.
Simona
Rota-Nodari
(Laboratoire Paul Painlevé, University of Lille, France)
Title : On a nonlinear Schrödinger
equation for nucleons.
Abstract: In this talk we consider
a model for a nucleon interacting with the σ and ω mesons
in the atomic nucleus. The model is relativistic, but we study it in
the nuclear physics nonrelativistic limit where is described by a
nonlinear Schrödinger-type equation with a mass which depends on the
solution itself.
After
discussing some previous results on the existence of positive
solutions, I will prove the uniqueness and non-degeneracy of these
ones.
The
talk is based on joint work with Mathieu Lewin
Julien
Sabin
(Département de Mathématiques, University of Orsay, France)
Title: Optimal trace ideals
properties of the restriction operator and applications
Abstract:
We study the trace ideals properties of the Fourier restriction
operator to hypersurfaces. Equivalently, we generalize the theorems of
Stein-Tomas and Strichartz to systems of orthonormal functions, with an
optimal dependence on the number of such functions. As an application,
we deduce new Strichartz inequalities describing the dispersive
behaviour of the free evolution of quantum systems with an infinite
number of particles. This is a joint work with Rupert Frank (Caltech).
Katharina
Schratz
(KIT, Karlsruhe Institut für Technologie, Karlsruhe, Germany)
Title: Efficient time integration
of Klein-Gordon type equations in high-frequency regimes
Abstract: The numerical
simulation
of the Klein-Gordon equation in the non-relativistic limit regime is
very delicate due to the highly oscillatory behavior of the solution.
In order to resolve the oscillations numerically, severe time step
restrictions need to be imposed, which leads to huge computational
efforts. In this talk we present an idea on the construction of
efficient robust numerical time integrators based on the asymptotic
expansion of the exact solution. This assymptotic approach allows us to
filter out the high oscillations in the exact solution explicitly and
the numerical task can be reduced to the simulation of the
corresponding non-oscillatory Schrödinger-type limit systems.
Hence, the computational costs can be drastically reduced. As this ansatz turns
out to be very promising in the second part of the talk we will give
some ideas how to extend the results to the Klein-Gordon-Zakharov
system.