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 the initial data u0 ∈ X, where X∈{M2,qs(ℝ),Hσ(T),Hs1(ℝ)+Hs2(T)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \in \{M_{2,q}^s(\mathbb{R}),\,{H^\sigma}(\mathbb{T}),\,{H^{{s_1}}}(\mathbb{R}) + {H^{{s_2}}}(\mathbb{T})\}$$\end{document} and q ∈ [1, 2], s ⩾ 0, or σ ⩾ 0, or s2 ⩾ s1 ⩾ 0. Moreover, if M2,qs(ℝ) ↪ L3(ℝ), or if σ⩾16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \geqslant {1 \over 6}$$\end{document}, or if s1⩾16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_1} \geqslant {1 \over 6}$$\end{document} and s2>12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s_2} > {1 \over 2}$$\end{document} we how that the Cauchy problem is unconditionally wellposed in X. Similar results hold true for all higher order nonlinear Schrödinger equations 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.