Nonlinear Schr\"odinger equation, differentiation by parts and modulation spaces

TL;DR

This paper establishes the existence and unconditional well-posedness of solutions for the cubic nonlinear Schrödinger equation in modulation spaces, extending previous results by applying a refined differentiation by parts technique for a broader range of parameters.

Contribution

It extends the differentiation by parts method to modulation spaces for the cubic NLS, covering cases with p ≠ 2 and providing new well-posedness results.

Findings

Existence of weak solutions in modulation spaces for certain p, q, s ranges.

Unconditional well-posedness in specified modulation spaces.

Improved estimates for p ≠ 2 using differentiation by parts.

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 $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and $s\geq0$. Moreover, for either $1\leq q\leq\frac32, s\geq0$ and $2\leq p\leq 3$ or $\frac32<q\leq\frac{18}{11}, s>\frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11}<q\leq2, s>\frac23-\frac1{q}$ and $2\leq p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}(\mathbb R).$ This improves \cite{NP}, 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 $p\neq2$.