Inhomogeneous nonlinear Schrödinger equation in Fourier-Lebesgue and modulation spaces

The purpose of this work is to provide a broader framework for analyzing the inhomogeneous nonlinear Schrödinger equation (INLS) $$i{u}_t+u_{xx}\pm \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -1}u=0,1<\alpha <5-2b\text{and}0<b\le 1\mathrm{/}4.$$ Specifically, we establish low-regularity well-posedness in the Fourier-Lebesgue ${\widehat{L}}^p$ spaces for $4\mathrm{/}3<p<8$. 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 $M^{p,p^{\mathrm{\prime }}}$ for $4\mathrm{/}3<p<2$.