论文标题
关于某些非线性schrödinger方程的自相似解决方案的存在
On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations
论文作者
论文摘要
我们构建了Schrödinger方程的解决方案，这些解决方案随着时间的流逝而具有渐近自我相似的解决方案。还包括带有两个bubbles的情况。这些解决方案是全局的，具有恒定的非零$ l^2 $规范，并且是稳定的。因此，它们不是线性波和局部波的标准渐近分解。鉴于先前关于通用分散方程的大时间行为的作品，预期这种弱的局部波是可以预期的。结果表明，可以将\ emph {散射通道}与这种溶液相关联，将扩张算子与渐近算子作为渐近的“哈密顿式”。
We construct solutions of Schrödinger 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".
