Global wellposedness of NLS in H1(R)+Hs(T)H^1(\mathbb{R})+H^s(\mathbb{T})

We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS [6] and global results for the quadratic NLS [8] in this hybrid setting

Unconditional uniqueness of higher order nonlinear Schrödinger equations

summary: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 $u_{0}\in X$, where $X\in \{M_{2,q}^{s}(\mathbb {R}), H^{\sigma }(\mathbb {T}), H^{s_{1}}(\mathbb {R})+H^{s_{2}}(\mathbb {T})\}$ and $q\in [1,2]$, $s\geq 0$, or $\sigma \geq 0$, or $s_{2}\geq s_{1}\geq 0$. Moreover, if $M_{2,q}^{s}(\mathbb {R})\hookrightarrow L^{3}(\mathbb {R})$, or if $\sigma \geq \frac 16$, or if $s_{1}\geq \frac 16$ and $s_{2}>\frac 12$ we show 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

Disputatio Politica Posterior, Disquirens: Num Iudex Semper Secundum Leges, An Etiam Interdum Secundum Aequitatem Iudicare Debeat?

Quam ... Praesidio M. Matthaei Kunstmann/ R. P. publico Doctorum Examini sistit Johannes Fridericus Kressen/ Regiomonte Pruss. Anno MDCLXXXVI. a. d. XI. Maii ..

Disputatio Politica Prior; Disquirens: Num Iudex Semper Secundum Leges, An Etiam Interdum Secundum Aequitatem Iudicare Debeat?

Quam ... Praeside M. Matthaeo Kunstmann/ R. P.Eruditorum publicae disceptationi subiicit Johannes Fridericus Kressen/ Regiomonte Pruss. Anno MDCLXXXVI. a. d. 8. Maii ..