Asymptotic stability of solitary waves for the 1D cubic-quintic Schrödinger equation with no internal mode

Abstract: For the Schr\"odinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.