Simplified title focusing on cubic nonlinear Schrödinger equation with rough potential

Simplified title focusing on cubic nonlinear Schrödinger equation with rough potential

This paper studies the cubic nonlinear Schrödinger equation with a spatially rough potential, which is key to modeling nonlinear Anderson localization where waves become localized under a rough potential. The work provides new optimal well-posedness analysis characterizing how the potential regularity impacts the solution regularity. Sharp ill-posedness results indicate the minimum potential regularity for solvability. A new efficient numerical method is proposed and analyzed, with simulations confirming the theoretical results and showcasing superior accuracy over existing methods.