On stability of small solitons of the 1--D NLS with a trapping delta potential

We consider a Nonlinear Schr"odinger Equation with a very general non linear term and with a trapping $\delta$ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. We give also a result of dispersion in the case of defocusing equations with a non–trapping delta potential.