Knocking out teeth in one-dimensional periodic NLS

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 u0 in H$^{s1}$(R)+H$^{s2}$(T),0≤s1≤s2. In addition, we show that if u0∈H$^{s}$(R)+H$^{1/2ϵ}$(T) where ϵ>0 and 1/6≤s≤1/2 the solution is unique in H$^{s}$(R)+H$^{1/2ϵ}$(T). Our main tool is a normal form type reduction via the use of the differentiation by parts technique.