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.