IDTPsi: Theory and Numerics

Laboratoire J. A. Dieudonné, Nice

January 12th-14th, 2015

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     Place: Salle de Conférences, Laboratoire J. A. Dieudonné.

Monday 12th
Tuesday 13th
Wednesday 14th




Ansgar Jüngel


Julien Sabin

Wolf Patrick Düll


Romain Duboscq





Christian Klein

Simona Rota-Nodari


Didier Clamond

Katharina Schratz

Rémi Carles


Laurent Di Menza

Titles and Abstracts :

Rémi Carles
(I3M, Institut de Mathématique et de Modélisation de Montpellier, University of Montpellier, France)

Title: Damped nonlinear Schrödinger equations

Abstract: We describe the effect of the introduction of a nonlinear damping in the dynamics of the Schrödinger equation. A superlinear damping prevents finite time blow-up if the damping term grows faster than the focusing nonlinearity at infinity. Moreover, in the presence of confinement (compact manifold or harominc potential), the global solution converges to zero for large time. On the other hand, introducing a sublinear damping in the linear Schrödinger equation leads to finite time extinction, in the presence of confinement. This talk is based upon joint works with Paolo Antonelli and Christof Sparber, Clément Gallo, Tohru Ozawa.

Didier Clamond
 (Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, France)

Title: Long time interaction of envelope solitons in the context of water waves.

Abstract: This presentation concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep
fluid in potential flow. Fully nonlinear simulations (Euler equations) show how an initially long wave group slowly splits into a number of solitary wave groups. This dynamics  is compared with simplified models (NLS, Dysthe, Zakharov, etc.) and some (un)expected behaviours will be discussed.

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)

Modeling and analysis of diffusive quantum fluid equations

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.