Asymptotic stability of travelling waves for general nonlinear Schrödinger equations with non-zero condition at infinity

In previous works [4, 5], existence and uniqueness of travelling waves for the nonlinear Schrödinger equations have been shown for speeds close to the speed of sound. Furthermore, it has been proved that a chain of dark solitons of well-ordered speeds near the sound speed, taken initially apart from each other, is orbitally stable. In this article, we complete this study by proving the asymptotic stability of these travelling waves, namely that a solution initially close to a travelling wave eventually converges towards a travelling wave of close speed. This relies on the methods used by F. Béthuel, P. Gravejat and D. Smets in [6] and first introduced by Y. Martel and F. Merle in [22].