Stability of the traveling waves for the derivative Schrödinger equation in the energy space

In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel–Merle–Tsai’s analytic approach in Martel et al. (Commun Math Phys 231(2):347–373, 2002, Duke Math J 133(3):405–466, 2006). As a by-product, we also give an alternative proof of the stability of the single traveling wave in the energy space in Colin and Ohta (Ann Inst Henri Poincaré Anal Non Linéaire 23(5):753–764, 2006), where Colin and Ohta made use of the concentration-compactness argument.