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

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 $M_{2,q}^s(\mathbb{R})$, $1\le q\le 2$ and $s\ge 0.$ In addition, for either $s\ge 0$ and $1\le q\le \frac{3}{2}$ or $\frac{3}{2}<q\le 2$ and $s>\frac{2}{3}-\frac{1}{q}$ 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.

Wrong statements and claims of the previous version have been fixed, unconditional wellposedness result has been added and the reference list has been expanded