Unconditional uniqueness of higher order nonlinear Schrödinger equations

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 $X\in \{M_{2,q}^s(\mathbb{R}),H^{\sigma }(\mathbb{T}),H^{s_1}(\mathbb{R})+H^{s_2}(\mathbb{T})\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s(\mathbb{R})\hookrightarrow L^3(\mathbb{R})$, 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 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

29 pages, authors list expanded, main theorems contain the general higher order NLS, reference list expanded, arXiv admin note: text overlap with arXiv:1802.08274, arXiv:1802.10464

References in corpus (4)

• Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrodinger equation
• Nonlinear Schrödinger equation, differentiation by parts and modulation spaces
• Unconditional uniqueness of solutions for nonlinear dispersive equations
• The global Cauchy problem for the NLS with higher order anisotropic dispersion