  In this paper, we prove the asymptotic stability of solitary waves to 1D
nonlinear Schr\"odinger equations in the subcritical case with symmetry and
spectrum assumptions. One of the main ideas is to use the vector fields method
developed by Cuccagna, Georgiev, Visciglia to overcome the weak decay with
respect to $t$ of the linearized equation caused by the one dimension setting
and the weak nonlinearity caused by the subcritical growth of the nonlinearity
term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of
solutions to 1D Schr\"odinger equations obtained by Staffilani to control the
high moments of the solutions emerging from the vector fields method.
