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Keywords:
normal form method; modulation space; unconditional uniqueness; higher order nonlinear Schrödinger
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.
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