# Finite difference scheme for two-dimensional periodic nonlinear Schr\"odinger equations

## TL;DR

This paper proves that a finite difference scheme for the two-dimensional periodic nonlinear Schrödinger equation converges strongly in L^2 to the true solution as the grid size decreases, validating the method's effectiveness.

## Contribution

It establishes the strong L^2 convergence of the finite difference scheme for 2D periodic NLS, providing a rigorous justification for its use.

## Key findings

- Strong L^2 convergence as grid size approaches zero
- Validation of finite difference method for 2D periodic NLS
- Theoretical proof of scheme's effectiveness

## Abstract

A nonlinear Schr\"odinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr\"odinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.