# A high-order nodal discontinuous Galerkin method for nonlinear fractional Schr\"{o}dinger type equations

## TL;DR

This paper introduces a high-order nodal discontinuous Galerkin method tailored for solving nonlinear fractional Schrödinger equations, demonstrating stability, optimal convergence, and verified through numerical experiments.

## Contribution

The paper develops a novel high-order discontinuous Galerkin method for nonlinear fractional Schrödinger equations, proving stability and convergence, with numerical validation.

## Key findings

- Proves $L^{2}$ stability for the proposed method.
- Achieves optimal convergence order of $O(h^{N+1})$.
- Numerical experiments confirm theoretical convergence rates.

## Abstract

We propose a nodal discontinuous Galerkin method for solving the nonlinear Riesz space fractional Schr\"{o}dinger equation and the strongly coupled nonlinear Riesz space fractional Schr\"{o}dinger equations. These problems have been expressed as a system of low order differential/integral equations. Moreover, we prove, for both problems, $L^{2}$ stability and optimal order of convergence $O(h^{N+1})$, where $h$ is space step size and $N$ is polynomial degree. Finally, the performed numerical experiments confirm the optimal order of convergence.