Optimal error bounds on an exponential wave integrator Fourier spectral method for fractional nonlinear Schrödinger equations with low regularity potential and nonlinearity

TL;DR

This work analyzes the space fractional nonlinear Schrödinger equation with rough potentials and nonlinearities by applying a first-order exponential wave integrator in time and a Fourier spectral method in space. It proves optimal $L^2$-error bounds for the semi-discrete scheme and optimal $L^2$- and $H^{\alpha/2}$-error bounds for the fully discrete scheme, with no CFL condition required, and shows improved energy-norm convergence under stronger regularity. The analysis encompasses low-regularity data via $V\in L^{\infty}$ and $f(\rho)=\beta\rho^{\sigma}$, with sharp dependence on the fractional order $\alpha$ and regularity $m$ of the exact solution. Numerical experiments validate the theory, demonstrate sharpness of the bounds, and reveal distinct dynamical patterns between SFNLSE and the classical NLSE, including how smaller $\alpha$ enhances localization and peak amplitudes.

Abstract

We establish optimal error bounds on an exponential wave integrator (EWI) for the space fractional nonlinear Schrödinger 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 $H^α$-solution with $1<α\leq 2$ being the fractional index of $(-Δ)^\fracα{2}$, we prove an optimal first-order $L^2$-norm error bound $O(τ)$ and a uniform $H^α$-norm bound of the semi-discrete numerical solution, where $τ$ 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 $O(τ+h^{m})$ 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 $O(τ)$ and $O(τ+h^{m-{\fracα{2}}})$ in $H^\fracα{2}$-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 Schrödinger equation.

Figures (10)

• Figure 1: Convergence tests of the EWI for \ref{['e5.1n']} with low regularity potential $V_1$ and smooth initial condition $\psi_0$ : $(a_{1})-(a_{3})$ spatial errors of the FS and FP discretizations, and $(b_{1})-(b_{3})$ temporal errors in $L^2$-norm and $H^{\frac{\alpha}{2}}$-norm.
• Figure 2: Convergence tests of the EWI for \ref{['e5.1n']} with low regularity potential $V_2$ and smooth initial condition $\psi_0$ : $(a_{1})-(a_{3})$ spatial errors of the FS and FP discretizations, and $(b_{1})-(b_{3})$ temporal errors in $L^2$-norm and $H^{\frac{\alpha}{2}}$-norm.
• Figure 3: Evolutions of $|\psi(x,t)|$ for the SFNLSE \ref{['e5.1n']} with low regularity potential $V_1$ and different fractional powers $\alpha=2, 1.5, 1.2$.
• Figure 4: Evolutions of $|\psi(x,t)|$ for the SFNLSE \ref{['e5.1n']} with low regularity potential $V_2$ and different fractional powers $\alpha=2, 1.5, 1.2$.
• Figure 5: Comparison of the FS and FP discretization of the nonlinear term in \ref{['e5.1']} with $\sigma=0.1$ and Type I initial datum \ref{['e5.2']}.

Theorems & Definitions (25)

• Remark 2.1: Implementation
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• proof