On Optimal Convergence Rates for the Nonlinear Schrödinger Equation with a Wave Operator via Localized Orthogonal Decomposition

In this paper, we develop a Localized Orthogonal Decomposition (LOD) method for the two-dimensional time-dependent nonlinear Schrödinger equation with a wave operator. We prove that our method preserves conservation laws and admits a unique numerical solution; furthermore, we obtain unconditional (i.e., time-step restriction-free) optimal-order superconvergent $L^p$ error estimates. To complement the theoretical analysis, we present a series of numerical simulations that verify the analytical results and further illustrate structural aspects of the problem.