Stability of small solitary waves for the 1$d$ NLS with an attractive delta potential

We consider the initial-value problem for the one-dimensional nonlinear Schr"odinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as $t\to \mathrm{\infty }$. In particular, we establish the asymptotic stability of the family of small solitary waves.