Optimal error bounds on the exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

TL;DR

This work addresses the nonlinear Schrödinger equation with $L^{\infty}$-potential and locally Lipschitz nonlinearity under $H^2$-solution assumptions. It develops and analyzes a first-order Gautschi-type exponential wave integrator (EWI) together with Fourier spectral (FS) and extended Fourier pseudospectral (EFP) spatial discretizations, proving optimal $L^2$-error $O(\tau)$ for semi-discretization and $O(\tau + h^2)$ for full discretization, without coupling between $\tau$ and $h$. Under higher regularity ($V\in W^{1,4}$ and $f$ meeting additional smoothness), $H^1$-error bounds are also established, and the EFP method extends these results when the potential is rough but the nonlinearity remains smooth. The authors show that EWI offers improved error bounds over classical methods in low-regularity settings and confirm sharpness through extensive numerical experiments, including comparisons with time-splitting schemes. The results enable reliable and efficient simulations of NLSE in physically relevant, non-smooth environments, such as Bose-Einstein condensates and nonlinear optics.

Abstract

We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with $ L^\infty $-potential and/or locally Lipschitz nonlinearity under the assumption of $ H^2 $-solution of the NLSE. For the semi-discretization in time by the first-order Gautschi-type EWI, we prove an optimal $ L^2 $-error bound at $ O(τ) $ with $ τ>0 $ being the time step size, together with a uniform $ H^2 $-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal $ L^2 $-error bound at $ O(τ+ h^2) $ without any coupling condition between $ τ$ and $ h $, where $ h>0 $ is the mesh size. In addition, for $ W^{1, 4} $-potential and a little stronger regularity of the nonlinearity, under the assumption of $ H^3 $-solution, we obtain an optimal $ H^1 $-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

Figures (8)

• Figure 1: Comparison of the Fourier spectral and pseudospectral discretization of the nonlinear term in \ref{['eq:NLSE_semi-smooth']} with $\sigma = 0.1$ and Type I initial datum \ref{['typeI_ini']}.
• Figure 2: Temporal errors of the EWI for the NLSE \ref{['eq:NLSE_semi-smooth']} with Type I initial datum \ref{['typeI_ini']}: (a) $L^2$-norm errors, and (b) $H^1$-norm errors.
• Figure 3: Comparison of the Fourier spectral and pseudospectral discretizations of the nonlinear term in \ref{['eq:NLSE_semi-smooth']} with $\sigma = 0.1$ and Type II initial datum \ref{['typeII_ini']}.
• Figure 4: Temporal errors of the EWI for the NLSE \ref{['eq:NLSE_semi-smooth']} with Type II initial datum \ref{['typeII_ini']}: (a) $L^2$-norm errors, and (b) $H^1$-norm errors.
• Figure 5: Convergence tests of the EWI for \ref{['eq:NLSE_Linfty_poten']} with $V=V_1 \in L^\infty(\Omega)$ and $\psi_0 \in H^2(\Omega)$: (a) spatial errors of the Fourier spectral and pseudospectral discretizations for the linear potential, and (b) temporal errors in $L^2$-norm and $H^1$-norm.

Theorems & Definitions (28)

• Theorem 3.1
• Remark 3.2
• Remark 3.3
• Lemma 3.4
• Proof 1
• Lemma 3.5
• Proof 2
• Lemma 3.6
• Proof 3
• Proposition 3.7: local truncation error