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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.

Asymptotic stability of solitons to 1D nonlinear Schrödinger equations in subcritical case