Stability of solitons for non-integrable NLS type equation with non-trivial far-field

In this presentation, devoted to the one-dimensional nonlinear Schrödinger equation with non-zero boundary conditions at infinity, we will investigate different notions of stability in order to study certain special solutions known as travelling waves. These solutions appear in the large-time dynamics of general dispersive systems, and we will explain how such solutions, referred to as solitons, are crucial for understanding the overall behavior of solutions to these equations and how they are related to the notion of integrability of the system. Many different behaviors for these travelling waves have been highlighted, according to the shape of the nonlinearity. Nevertheless, we have been able to prove the existence of travelling waves with small momentum. Moreover, we shall dwell on the existence and uniqueness travelling waves with speed close to the speed of sound, the orbital stability of a well-prepared chain of such travelling waves, as well as the asymptotic stability of these special solutions.