标题： 一维非线性薛定谔方程孤立波的渐近稳定性 显示英文标题

标题： Asymptotic Stability of Solitary Waves for One Dimensional Nonlinear Schrödinger Equations

摘要： 我们证明了非线性薛定谔方程在空间维数1时孤立波的全局渐近稳定性。 此外，如果非线性在原点处为三次方，则辐射表现出长程散射；如果非线性为更高阶，则表现出标准散射。 我们处理了一种一般的非线性，没有任何消失条件，要求围绕孤立波的线性化算子既没有非零特征值，也没有临界共振。 初始数据在自然空间$H^1 \cap L^{2,1}$中孤立波的邻域内选取（后者是加权$L^2$空间）。 证明依赖于通过变形傅里叶变换观察到的共振分析，并首次结合了调制和重正化技术。 显示英文摘要

摘要： We show global asymptotic stability of solitary waves of the nonlinear Schr\"odinger 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, 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 the weighted $L^2$ space). The proof relies on the analysis of resonances as seen through the distorted Fourier transform, combined for the first time with modulation and renormalization techniques.