Geodesic
Unconditional uniqueness of higher order nonlinear Schrödinger equations
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 709-742
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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.
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.
DOI : 10.21136/CMJ.2021.0078-20
Classification : 35A01, 35A02, 35D30, 35J30
Keywords: normal form method; modulation space; unconditional uniqueness; higher order nonlinear Schrödinger 
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Klaus, Friedrich; Kunstmann, Peer; Pattakos, Nikolaos. Unconditional uniqueness of higher order nonlinear Schrödinger equations. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 3, pp. 709-742. doi: 10.21136/CMJ.2021.0078-20

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