I will present a proof of the asymptotic stability of solitary waves for the 1D cubic NLS under even perturbations based on a combination of modulation techniques and a space-time resonances approach using the distorted Fourier transform. The main challenge are the threshold resonances of the linearized operator and the resulting slow local decay of the Schrödinger waves. Remarkable null structures in the evolution equation for the radiation term as well as in the modulation equations play an important role in the proof. This is joint work with Yongming Li (Texas A&M University).