NLS in the modulation space $M_{2,q}(\mathbb R)$

TL;DR

This paper proves the existence and unconditional well-posedness of solutions for the cubic nonlinear Schrödinger equation in certain modulation spaces, extending previous techniques from periodic to non-periodic settings.

Contribution

It establishes weak solutions and unconditional well-posedness of NLS in modulation spaces $M_{2,q}^s(R)$ for specific ranges of $q$ and $s$, using differentiation by parts.

Findings

Existence of weak solutions in $M_{2,q}^s(R)$ for $1 extless q extless 2$, $s extgreater 0$.

Unconditional well-posedness in $M_{2,q}^s(R)$ for specified $q$ and $s$ ranges.

Application of differentiation by parts technique in the non-periodic setting.

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the modulation space $M_{2,q}^{s}(\mathbb R)$, $1\leq q\leq2$ and $s\geq0.$ In addition, for either $s\geq 0$ and $1\leq q\leq\frac32$ or $\frac32<q\leq 2$ and $s>\frac23-\frac1{q}$ we show that the Cauchy problem is unconditionally wellposed in $M_{2,q}^{s}(\mathbb R).$ It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.