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

TL;DR

This paper establishes optimal error bounds for an exponential wave integrator Fourier spectral method applied to fractional nonlinear Schrödinger equations with low regularity potential and nonlinearity, demonstrating high accuracy without CFL restrictions.

Contribution

The paper introduces and analyzes an optimal error bound for a Fourier spectral method combined with an exponential wave integrator for SFNLSE with low regularity data, extending previous results.

Findings

Optimal first-order $L^2$-norm error bound $O( au)$ for semi-discretization.

Error bound $O( au+h^{m})$ in $L^{2}$-norm without CFL restrictions.

Distinct evolving patterns between SFNLSE and classical nonlinear Schrödinger equation.

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.