为什么孤子是稳定的？
Why are solitons stable?
摘要：
线性色散方程理论预测，波应该随着时间的推移而扩散和色散。然而，在理论和实践中观察到的一个显著现象是，一旦考虑非线性效应，就可以创建\emph{solitary wave}或\emph{soliton}解，其可以足够稳定以无限期地持续。这种解的构造相对简单，但它们是\emph{stable}这一事实需要大量的分析才能建立，部分原因是方程中的对称性（如平移不变性）在稳定性分析中产生了退化。该理论在\emph{critical}的情况下是特别困难的，在这种情况下，非线性是在正确的权力，以潜在地允许自相似爆破。在这篇文章中，我们调查的一些亮点，从温斯坦和Grillakis-Shatah-Strauss更经典的轨道稳定性分析，最近的渐近稳定性和爆破分析的Martel-Merle和Merle-Raphael，以及目前的发展，在使用这个理论严格证明控制爆破的几个关键方程。
英文摘要：
The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \emph{solitary wave} or \emph{soliton} solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions is relatively straightforward, but the fact that they are \emph{stable} requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the \emph{critical} case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.
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