ReviewA high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations

- A nodal discontinuous Galerkin method for solving the nonlinear fractional Schrödinger equation and the strongly coupled nonlinear fractional Schrödinger equations has been proposed.
- The performed numerical experiments confirm the optimal order of convergence.
- When order of fractional derivative tends to 2, the shape of the solitons will change more slightly and the waveforms become closer to the classical case.

We propose a nodal discontinuous Galerkin method for solving the nonlinear Riesz space fractional Schrödinger equation and the strongly coupled nonlinear Riesz space fractional Schrödinger equations. These problems have been expressed as a system of low order differential/integral equations. Moreover, we prove, for both problems, L2 stability and optimal order of convergence O(hN+1), where h is space step size and N is polynomial degree. Finally, the performed numerical experiments confirm the optimal order of convergence.