Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation

For the Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity in one space dimension, this article proves the local asymptotic completeness of the family of small standing solitary waves under even perturbations in the energy space. For this model, perturbative of the integrable cubic Schrödinger equation for small solutions, the linearized equation around a small solitary wave has an internal mode, whose contribution to the dynamics is handled by the Fermi golden rule.