  We consider perturbations of the one-dimensional cubic Schr\"odinger
equation, under the form $i \, \partial_t \psi + \partial_x^2 \psi + |\psi|^2
\psi - g( |\psi|^2 ) \psi = 0$. 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.
