Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

Abstract

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{p,q}^{s}({\mathbb {R}})$ where $1\le q\le 2$, $2\le p<\frac{10q'}{q'+6}$ and $s\ge 0$. Moreover, for either $1\le q\le \frac{3}{2}, s\ge 0$ and $2\le p\le 3$ or $\frac{3}{2}<q\le \frac{18}{11}, s>\frac{2}{3}-\frac{1}{q}$ and $2\le p\le 3$ or $\frac{18}{11}<q\le 2, s>\frac{2}{3}-\frac{1}{q}$ and $2\le p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}({\mathbb R}).$ This improves Pattakos (J Fourier Anal Appl, 2018. https://doi.org/10.1007/s00041-018-09655-9), where the case $p=2$ was considered and the differentiation-by-parts technique was introduced to a problem with continuous Fourier variable. Here, the same technique is used, but more delicate estimates are necessary for $p\ne 2$.