Czechoslovak Mathematical Journal, Vol. 71, No. 3, pp. 709-742, 2021
Unconditional uniqueness of higher order nonlinear Schrödinger equations
Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos
Received February 25, 2020. Published online April 20, 2021.
Abstract: 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 $q\in[1,2]$, $s\geq0$, or $\sigma\geq0$, or $s_2\geq s_1\geq0$. Moreover, if $M_{2,q}^s(\mathbb{R})\hookrightarrow L^3(\mathbb{R})$, or if $\sigma\geq\frac16$, or if $s_1\geq\frac16$ and $s_2>\frac12$ 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.
Keywords: normal form method; modulation space; unconditional uniqueness; higher order nonlinear Schrödinger
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Affiliations: Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (corresponding author), Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology (KIT), Englerstrasse 2, 76128 Karlsruhe, Germany, e-mail: friedrich.klaus@kit.edu, peer.kunstmann@kit.edu, nikolaos.pattakos@kit.edu
