Localization of solitons: linear response of the mean-field ground state to weak external potentials

Two aspects of bright matter-wave solitons in weak external potentials are discussed. First, we briefly review recent results on the Anderson localization of an entire soliton in disordered potentials [Sacha et al. PRL 103, 210402 (2009)], as a paradigmatic showcase of genuine quantum dynamics beyond simple perturbation theory. Second, we calculate the linear response of the mean-field soliton shape to a weak, but otherwise arbitrary external potential, with a detailed application to lattice potentials.