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

We consider perturbations of the one-dimensional cubic Schr\"odinger equation, of the form i \, ∂ₜ ψ + ∂ₓ² ψ + |ψ|² ψ + g( |ψ|² ) ψ = 0. Under hypotheses on the function g that can be easily verified in some cases (such as g(s) = s^σ with σ >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^σ, we prove the asymptotic stability of the solitary waves with small frequencies.