Asymptotic stability of solitons for near-cubic NLS equation with an internal mode

We consider perturbations of the one-dimensional cubic Schrödinger equation, of 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 (such as $g(s)=s^{\sigma }$ with $\sigma >1$), we show that the linearized problem around a small solitary wave presents a unique internal mode. Moreover, under an additional hypothesis (the Fermi golden rule) that can also be verified in the case of powers $g(s)=s^{\sigma }$, we prove the asymptotic stability of the solitary waves with small frequencies.