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Unconditional uniqueness of higher order nonlinear Schrödinger equations

Kunstmann, Peer; Pattakos, Nikolaos

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 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.

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Volltext §
DOI: 10.5445/IR/1000099946
Veröffentlicht am 18.11.2019
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Analysis (IANA)
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2019
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000099946
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 28 S.
Serie CRC Preprint ; 2019/22
Projektinformation SFB 1173/2 (DFG, DFG KOORD, SFB 1173/2 2019)
Externe Relationen Siehe auch
Schlagwörter Normal form method, modulation spaces, unconditional uniqueness, higher order nonlinear Schrödinger
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