Inhomogeneous nonlinear Schr\"odinger equation in Fourier-Lebesgue and modulation spaces

Abstract

The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schr\"odinger equation (INLS) \[iu_t + u_{xx} \pm |x|^{-b}|u|^{\alpha-1}u=0, \quad 1<\alpha<5-2b\; \text{and}\; 0<b\leq 1/4.\] Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue $\widehat{L}^{p}$ spaces for $4/3<p<8$. The analysis is carried out differently for the cases $p<2$ and $p>2$. Primarily, in both cases, we prove global well-posedness for arbitrarily large initial data via the data decomposition method adapted for the Fourier-Lebesgue spaces. Furthermore, we obtain analogous results for INLS in modulation spaces $M^{p,p'}$ for $4/3<p<2$.