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.

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

References in corpus (1)

• Local well-posedness of nonlinear dispersive equations on modulation spaces

Cited by in corpus (5)

• Nonlinear Schrödinger equation, differentiation by parts and modulation spaces
• Knocking out teeth in one-dimensional periodic NLS
• Well-posedness for the NLS hierarchy
• Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces $M_{p,q}^s({\mathbb{R}}^d)\cap M_{\infty ,1}({\mathbb{R}}^d)$
• Global well-posedness and equicontinuity for mKdV in modulation spaces