Neba Polneau (doctorant IMJ-SU) - Equations de Schrödinger non-linéaires en une dimension d’espace

Schrödinger equations belong to the family of dispersive equations, characterized by the fact that wave packets propagate at speeds dependent on their frequency. This dispersion phenomenon has profound effects on the time behavior of the solutions and will be the central theme of this presentation. In the first part, we will focus on the linear case. We will explain the dispersion mechanism and its consequences for the time decay of the solutions, highlighting the fundamental estimates that follow from it. In the second part, we will move on to nonlinear models. Time permitting, we will study the long-time behavior of "small" solutions whose initial data belong to a weighted Sobolev space. We will introduce the notion of scattering for nonlinear partial differential equations, which essentially describes the fact that a nonlinear solution behaves asymptotically like a linear solution.