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In this paper, we analyze the long-time dynamics of small solutions to the$1d$ cubic nonlinear Schr\"odinger equation (NLS) with a trapping potential. Weshow that every small solution will decompose into a small solitary wave and aradiation term which exhibits the modified scattering. In particular, thisresult implies the asymptotic stability of small solitary waves. Our analysisalso establishes the long-time behavior of solutions to a perturbation of theintegrable cubic NLS with the appearance of solitons.

Long-time dynamics of small solutions to 1$d$ cubic nonlinear Schr\"odinger equations with a trapping potential