Asymptotic stability of solitary waves for the 1D near-cubic non-linear Schrödinger equation in the absence of internal modes

We consider perturbations of the one-dimensional cubic Schrödinger equation, under the form $i{\mathrm{\partial }}_t\psi +{\mathrm{\partial }}_x^2\psi +\mathrm{\mid }\psi {\mathrm{\mid }}^2\psi -g(\mathrm{\mid }\psi {\mathrm{\mid }}^2)\psi =0$. Under hypotheses on the function g that can be easily verified in some cases, we show that the linearized problem around a solitary wave does not have internal mode (nor resonance) and we prove the asymptotic stability of these solitary waves, for small frequencies.