Unconditional uniqueness of higher order nonlinear Schr\"odinger equations

TL;DR

This paper establishes the unconditional well-posedness and existence of weak solutions for higher order nonlinear Schrödinger equations with various initial data spaces, using normal form reduction and phase factor analysis.

Contribution

It extends well-posedness results to higher order NLS equations and mixed orders, employing a novel factorization approach and differentiation by parts technique.

Findings

Existence of weak solutions in extended data spaces.

Unconditional well-posedness under specific embedding conditions.

Applicability to all 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 Schr\"odinger 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 $q\in[1,2]$, $s\geq0$, 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 Schr\"odinger 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