Brahim Alouini (University of Monastir): Asymptotic dynamics of solutions for a Nonlinear Schrödinger equation with energy critical nonlinear damping

Recent experiments have revealed the formation of stable droplets in a dipolar Bose-Einstein condensate (BEC). This surprising result has been explained, experimentally, by the stabilization given by the three body loss process appearing in the form of a critical nonlinear damping term in a nonlinear Schrödinger equation modelling the formation of these dipolar quantum droplets. The purpose of the current talk is to study the dynamics of solutions to this equation. In the first part, we present a mathematical study that validates this prediction by proving that the nonlinear damping prevents the collapse and ensures the existence of global-in-time solutions. The asymptotic dynamics of these solutions will be discussed as well, especially when the system is free (without potential). We show that all global solutions behave as free waves asymptotically in time.