论文标题
1-D中的立方NLS及其稳定性的多层
Multisolitons for the cubic NLS in 1-d and their stability
论文作者
论文摘要
对于立方非线性schrödinger方程（NLS）以及 修改的Korteweg-de Vries（MKDV）方程在一个空间维度中，我们考虑了纯$ n $ soliton States的集合$ {\ bf m} _n $及其相关的多层解决方案。我们证明（i）集合$ {\ bf m} _n $是一个均匀平滑的流形，（ii）$ {\ bf m} _n $状态在$ h^s $中均匀地稳定，对于每种$ s> - \ frac12 $。 我们分析中的一个主要工具是迭代的反弹变换，它使我们可以非线性地将多层添加到现有的无孤子自由状态（Soliton添加映射），或者可以从多层状态（Soliton Removal Map）中删除Multisoliton。这些地图的特性和规律性进行了广泛的研究。
For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set ${\bf M}_N$ of pure $N$-soliton states, and their associated multisoliton solutions. We prove that (i) the set ${\bf M}_N$ is a uniformly smooth manifold, and (ii) the ${\bf M}_N$ states are uniformly stable in $H^s$, for each $s>-\frac12$. One main tool in our analysis is an iterated Backlund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.
