## Section:
Scientific Foundations2>
### Numerical analysis of Schrödinger equations3>

Dispersive equations, Schrödinger equations

#### Modelling of quantum dot-helium4>

In collaboration with G. Reinish (Nice Observatory)
and V. Guðmundsson (University of Reykjavik), C. Besse and G. Dujardin are working on the numerical computation of the ground state and the first bound states of the non linear Schrödinger-Poisson system with confining quadratic potential in 2 space dimensions. This models quatum dot helium (*i.e.* the behavior of a pair of quantum electrons in a strong confining potential).
The goal is to perform after that numerical time stepping methods to simulate the dynamics of the NLSP system and compute accurately some quantities of physical interest as functions of time, in order to be able to compare the competition
between the Coulomb (repulsive) interaction and the binding (attractive)
forces due to the confinement in this model as well as in other quantum
mechanics models.

#### Dispersive Schrödinger-like equations4>

In collaboration with M. Taki (PhLAM laboratory, Lille), C. Besse and G. Dujardin are considering dispersive equations modelling the propagation of a laser beam in an optical fiber. They are trying to explain the possible ways of creating rogue waves in the propagation of laser beams. More generally, they are trying to explain which terms in the dispersive Schrödinger-like equations obtained by the physicists allow which physical behaviour of the solutions (e.g. the creation of rogue waves).