Knocking out teeth in one-dimensional periodic NLS

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger 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 $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T)$ where $\epsilon>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+\epsilon}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.