Optimal error bounds on an exponential wave integrator Fourier spectral method for fractional nonlinear Schr\"{o}dinger equations with low regularity potential and nonlinearity

Abstract

We establish optimal error bounds on an exponential wave integrator (EWI) for the space fractional nonlinear Schr\"{o}dinger equation (SFNLSE) with low regularity potential and/or nonlinearity. For the semi-discretization in time, under the assumption of $L^\infty$-potential, $C^1$-nonlinearity, and $H^\alpha$-solution with $1<\alpha \leq 2$ being the fractional index of $(-\Delta)^\frac{\alpha}{2}$, we prove an optimal first-order $L^2$-norm error bound $O(\tau)$ and a uniform $H^\alpha$-norm bound of the semi-discrete numerical solution, where $\tau$ is the time step size. We further discretize the EWI in space by the Fourier spectral method and obtain an optimal error bound in $L^{2}$-norm $O(\tau+h^{m})$ without introducing any CFL-type time step size restrictions, where $h$ is the spatial step size, $m$ is the regularity of the exact solution. Moreover, under slightly stronger regularity assumptions, we obtain optimal error bounds $O(\tau)$ and $O(\tau+h^{m-{\frac{\alpha}{2}}})$ in $H^\frac{\alpha}{2}$-norm, which is the norm associated to the energy. Extensive numerical examples are provided to validate the optimal error bounds and show their sharpness. We also find distinct evolving patterns between the SFNLSE and the classical nonlinear Schr\"{o}dinger equation.