一维周期性NLS中的打齿
Knocking out teeth in one-dimensional periodic NLS
摘要：
我们证明了一维三次非线性薛定谔方程的Cauchy问题在扩展意义下的弱解的存在性，初始数据为$u_{0}$在$H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$此外，我们还证明了如果$u_{0}\in H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T)$，其中$ε>0$和$\frac16\leq s\leq\frac12$，则解在$H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T).$中是唯一的。我们的主要工具是通过使用分部微分技术进行规范型约化。
英文摘要：
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T)$ where $ε>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.
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