An explicit and symmetric exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

TL;DR

This work develops a second-order explicit and time-symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation with low regularity potential and nonlinearities. It provides rigorous error estimates across regimes of potential and nonlinearity regularity, including optimal $L^2$-convergence for good data and first-order convergence under low-regularity, with an improved bound enabled by a regularity compensation oscillation (RCO) technique for certain non-resonant time steps. The method remains explicit and stable under a time-step restriction independent of the spatial mesh when combined with a Fourier spectral spatial discretization, and it demonstrates strong long-time behavior with near conservation of mass and energy. Extensive numerical experiments corroborate the theoretical results and highlight the sEWI’s robustness to rough potentials/nonlinearities and its efficiency in higher dimensions. Overall, the sEWI advances high-order, structure-preserving simulation for NLSEs in rough media, reducing regularity requirements while maintaining accuracy and stability.

Abstract

We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ |ψ|^{2σ}ψ$ with $ ψ$ being the wave function and $ σ> 0 $ being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For ``good" potential and nonlinearity ($H^2$-potential and $σ\geq 1$), we establish an optimal second-order error bound in the $L^2$-norm. For low regularity potential and nonlinearity ($L^\infty$-potential and $σ> 0$), we obtain a first-order $L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of the numerical solution. Moreover, adopting a new technique of \textit{regularity compensation oscillation} (RCO) to analyze error cancellation, for some non-resonant time steps, the optimal second-order $L^2$-norm error bound is proved under a weaker assumption on the nonlinearity: $σ\geq 1/2$. For all the cases, we also present corresponding fractional order error bounds in the $H^1$-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.

Figures (6)

• Figure 1: $L^2$- and $H^1$-errors of the sEWI for the NLSE with $\sigma = 1.1$ and $V \in H^2$ given in \ref{['eq:good']}: (a) temporal errors and (b) spatial errors.
• Figure 2: $L^2$- and $H^1$-errors of the sEWI for the NLSE with $\sigma = 0.1$ and $V \in L^\infty$ given in \ref{['eq:lowreg']}: (a) temporal errors and (b) spatial errors.
• Figure 3: $L^2$- and $H^1$-errors of the sEWI for the NLSE with $\sigma = 0.5$: (a) errors computed with $h = \sqrt{10\tau}$ and (b) errors computed with $h = w^{-9}$ fixed.
• Figure 4: Relative errors of (a) mass and (b) energy of the sEWI for the NLSE with $\sigma = 1.1$ and $V \in H^2$ given in \ref{['eq:good']}.
• Figure 5: Relative errors of (a) mass and (b) energy of the sEWI for the NLSE with $\sigma = 0.1$ and $V \in L^\infty$ given in \ref{['eq:V_low_1D']}.

Theorems & Definitions (38)

• Remark 2.1
• Theorem 3.1: Optimal error bounds for "good" potential and nonlinearity
• Theorem 3.2: Error bounds for low regularity potential and/or nonlinearity
• Remark 3.3
• Remark 3.4
• Lemma 3.5
• Lemma 3.6
• Proposition 3.7
• Proposition 3.8
• Proof 1