Unconditional uniqueness of higher order nonlinear Schrödinger equations

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with initial data $u_0\in X$, where $X\in\{M^s_{2,q}(\mathbb{R}), H^\sigma(\mathbb{T}), H^{s_1}(\mathbb{R})+H^{s_2(\mathbb{T})}\}$ and $q\in[1,2]$, $s\ge0$, $\sigma\ge0$, or $s_2\ge s_1\ge0$. Moreover, if $M^s_{2,q}(\mathbb{R})\hookrightarrow L^3(\mathbb{R})$, or if $\sigma\ge\frac{1}{6}$ or if $s_1\ge\frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for the cubic sixth order nonlinear Schrödinger equation and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ
the normal form reduction via the differentiation by parts technique and build upon our previous work.