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Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Rémi Carles, Bijan Mohammadi (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple...

Résonances de Rayleigh en dimension 2

Didier Gamblin (2004)

Bulletin de la Société Mathématique de France

Nous étudions les résonances de Rayleigh créées par un obstacle strictement convexe à bord analytique en dimension 2. Nous montrons qu’il existe exactement deux suites de résonances ( z k , + ) et ( z k , - ) convergeant exponentiellement vite vers l’axe réel dans un voisinage polynomial de l’axe réel, et exponentiellement proches d’une suite de quasimodes réels. De plus, k - 1 ℜ z k , ± est un symbole analytique d’ordre 0 en la variable k - 1 dont on donne le premier terme du développement. Nous construisons pour cela des quasimodes...

Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator

Bernard Helffer, Thierry Ramond (2000)

Journées équations aux dérivées partielles

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called “transfer operator method”. We are interested here in the thermodynamical limit Λ ( h ) of the ground state energy of this operator. For Kac’s spin model, Λ ( h ) is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied...

Semiclassical Limit of the cubic nonlinear Schrödinger Equation concerning a superfluid passing an obstacle

Fanghua Lin, Ping Zhang (2004/2005)

Séminaire Équations aux dérivées partielles

In this paper, we study the semiclassical limit of the cubic nonlinear Schrödinger equation with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches 0 .

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