标题： 关于某些非线性薛定谔方程自相似解的存在性 显示英文标题

标题： On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations

摘要： 我们构造了薛定谔方程的解，这些解在时间趋于无穷时具有渐近自相似解。 还包括双泡状的情况。 这些解是全局的，具有非零常数的$L^2$范数，并且是稳定的。 因此，它们不属于线性波和局域波的标准渐近分解。 考虑到一般色散方程的大时间行为的先前工作，这类弱局域波是预期的。 结果显示，可以将一个\emph{散射通道}与这些解相关联，其中缩放算子作为渐近的“哈密顿量”。 显示英文摘要

摘要： We construct solutions of Schr\"odinger equations which have asymptotic self similar solutions as time goes to infinity. Also included are situations with two-bubbles. These solutions are global, with constant non-zero $L^2$ norm, and are stable. As such they are not of the standard asymptotic decomposition of linear wave and localized waves. Such weakly localized waves were expected in view of previous works on the large time behavior of general dispersive equations. It is shown that one can associate a \emph{scattering channel} to such solutions, with the Dilation operator as the asymptotic "hamiltonian".