We consider the cubic nonlinear Schr\"odinger equation with a spatially rough potential, a key equation in the mathematical setup for nonlinear Anderson localization. Our study comprises two main parts: new optimal results on the well-posedness analysis on the PDE level, and subsequently a new efficient numerical method, its convergence analysis and simulations that illustrate our analytical results. In the analysis part, our results focus on understanding how the regularity of the solution is influenced by the regularity of the potential, where we provide quantitative and explicit characterizations. Ill-posedness results are also established to demonstrate the sharpness of the obtained regularity characterizations and to indicate the minimum regularity required from the potential for the NLS to be solvable. Building upon the obtained regularity results, we design an appropriate numerical discretization for the model and establish its convergence with an optimal error bound. The numerical experiments in the end not only verify the theoretical regularity results, but also confirm the established convergence rate of the proposed scheme. Additionally, a comparison with other existing schemes is conducted to demonstrate the better accuracy of our new scheme in the case of a rough potential.

翻译：本文研究具有空间粗糙势的三次非线性薛定谔方程，该方程是非线性安德森局域化数学框架中的关键方程。我们的研究包含两个主要部分：在偏微分方程层面上关于适定性分析的新最优结果，以及随后提出的新型高效数值方法及其收敛性分析与仿真验证。在分析部分，我们重点探究势函数正则性如何影响解的正则性，并给出定量显式刻画。为证明所获正则性刻画的精确性并指明方程可解所需势函数的最小正则性，我们还建立了不适定性结果。基于所得正则性结论，我们设计了该模型合适的数值离散格式，并证明其收敛性与最优误差界。最后的数值实验不仅验证了理论正则性结果，还确认了所提格式的收敛阶。此外，通过与其他现有格式的比较，证明了新格式在粗糙势情形下具有更优的精度。