We investigate numerically the behavior of a particle, or a set of particles, which exchange momentum and energy with the environment, described as a transverse vibrational field. The large time behavior is characterized, in some specific circumstances, by means of effective friction force and Landau damping. In order to discuss these issues on numerical grounds, we set up a dedicated numerical method. The scheme couples a Finite Element Method for the wave equation, with an appropriate transparent boundary condition that preserves the dispersive effects driving the asymptotic behavior of the system, and a symplectic scheme or a Semi-Lagrangian method for the evolution of the particles. The time discretisation is constructed to capture accurately the energy exchanges. The numerical simulations shed new light on the theoretical results and bring out clearly the role of the parameters of the models.

We investigate the existence and stability of ground states for a model coupling the Schrödinger equation to the wave equation in transverse directions. The model is intended to describe complex interactions between quantum particles and their environment. The result can be interpreted as a dissipation statement, induced by the energy exchanges with the environment. The proofs use either concentration-compactness arguments or spectral analysis of the linearized energy. Difficulties arise related to the fact the model does not satisfy scale invariance properties.Résumé (Sur les systèmes quantiques dissipatifs : états fondamentaux et stabilité orbitale)Nous étudions l'existence et la stabilité des états fondamentaux pour un modèle couplant l'équation de Schrödinger à l'équation d'onde dans des directions transverses. Ce modèle vise à décrire les interactions complexes entre des particules quantiques et leur environnement. Le résultat peut être interprété comme une propriété de dissipation, induite par les échanges d'énergie avec l'environnement. Les démonstrations reposent soit sur des arguments de concentrationcompacité, soit sur une analyse spectrale de l'énergie linéarisée. Des difficultés surviennent liées au fait que le modèle ne satisfait pas de propriétés d'invariance d'échelle.

We consider a simple model describing the interaction of a quantum particle with a vibrational environment, which eventually acts as a friction on the particle. This equation admits soliton-like solutions, and we numerically investigate their stability when subjected to a small initial impulsion. Our findings illustrate the analogies with the behavior of classical particles and the relevance of asymptotic models.