Asymptotic stability of solitary waves for one-dimensional nonlinear Schrödinger equations

We show global asymptotic stability of solitary waves of the nonlinear Schrödinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard scattering if it is higher order. We handle a general nonlinearity without any vanishing condition, but requiring that the linearized operator around the solitary wave has neither nonzero eigenvalues, nor threshold resonances. Initial data are chosen in a neighborhood of the solitary waves in the natural space H^{1} \cap L^{2,1} (where the latter is a weighted L^{2} space). The proof combines for the first time modulation and renormalization techniques with the distorted Fourier transform.