三次Wave Equation的自相似解
Self-similar Solutions of the Cubic Wave Equation
摘要：
我们证明了三维空间中的聚焦立方波方程存在可数族自相似解，这些解在奇点的过去光锥内是光滑的。这些解决方案是由一个整数索引$n$计数的解决方案的振荡的数量标记。的线性算子周围的$n$ -th解决方案被证明有$n+1$负本征值（其中之一对应于规范模式），这意味着所有的$n>0$解决方案是不稳定的。它还表明，所有的$n>0$解决方案有一个奇异性以外的过去的光锥，怀疑这些解决方案是否可能参与柯西演化，即使是非通用的初始数据。
英文摘要：
We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$ which counts the number of oscillations of the solution. The linearized operator around the $n$-th solution is shown to have $n+1$ negative eigenvalues (one of which corresponds to the gauge mode) which implies that all $n>0$ solutions are unstable. It is also shown that all $n>0$ solutions have a singularity outside the past light cone which casts doubt on whether these solutions may participate in the Cauchy evolution, even for non-generic initial data.
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