Orbital stability of a chain of dark solitons for general non-integrable Schrödinger equations with non-zero condition at infinity

In this article, we focus on the stability of dark solitons for a general one-dimensional nonlinear Schrödinger equation. More precisely, we prove the orbital stability of a chain of well-prepared traveling waves whose speeds are well ordered, taken close to the speed of sound c s and such that the solitons are initially localized far away from each other. The proof relies on the arguments developed by Béthuel et al (2014 Ann. Inst. Fourier 64 19–70) and first introduced in Martel et al (2002 Commun. Math. Phys. 231 347–73).