Explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation

TL;DR

The paper tackles numerical integration of the cubic nonlinear Schrödinger equation with low-regularity data by constructing the first fully explicit symmetric low-regularity integrators (sLRIs). It develops a general two-step framework that converts explicit one-step flows into symmetric two-step schemes, preserves structure, and achieves unconditional stability under suitable assumptions. The authors provide concrete explicit schemes (sLRI1 and sLRI2), extend the approach to ultra-low regularity and higher-order schemes via decorated trees, and establish rigorous convergence results including fractional gains in regularity due to symmetry. Numerical experiments in one dimension confirm the expected convergence, demonstrate near conservation of mass and energy over long times, and show computational advantages over implicit symmetric LRIs. The work broadens the toolkit for dispersive PDEs, enabling efficient, structure-preserving simulations at low regularity and suggesting applicability to a wider class of equations beyond the NLSE.

Abstract

The numerical approximation of low-regularity solutions to the nonlinear Schrödinger equation is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing symmetric low-regularity integrators for this equation. However, so far, all prior symmetric low-regularity algorithms are fully implicit, and therefore require the solution of a nonlinear equation at each time step, leading to significant numerical cost in the iteration. In this work, we introduce the first fully explicit (multi-step) symmetric low-regularity integrators for the nonlinear Schrödinger equation. We demonstrate the construction of an entire class of such schemes which notably can be used to symmetrise (in explicit form) a large amount of existing low-regularity integrators. We provide rigorous convergence analysis of our schemes and numerical examples demonstrating both the favourable structure preservation properties obtained with our novel schemes, and the significant reduction in computational cost over implicit methods.

Figures (11)

• Figure 1: Errors in $L^2$- and $H^1$-norms of the explicit sLRI1 for the NLSE \ref{['NLSE']} with initial data of regularity $H^\alpha \ (\alpha = 1, 2, 3, 4)$
• Figure 2: Errors in $L^2$- and $H^1$-norms of the explicit sLRI2 for the NLSE \ref{['NLSE']} with initial data of regularity $H^\alpha \ (\alpha = 1, 1.5, {2.25}, 3)$
• Figure 3: Relative errors of mass (left) and energy (right) of different methods for the NLSE \ref{['NLSE']} with $H^{1}$ initial datum
• Figure 4: Relative errors of mass (left) and energy (right) of different methods for the NLSE \ref{['NLSE']} with $H^{1.5}$ initial datum
• Figure 5: Relative errors of mass (left) and energy (right) of different methods for the NLSE \ref{['NLSE']} with $H^{2}$ initial datum

Theorems & Definitions (30)

• Definition 2.1
• Remark 2.3
• Remark 2.4
• Proposition 2.5
• Proof 1
• Proposition 3.1: Stability
• Proof 2
• Proposition 3.2: Local error
• Proof 3
• Theorem 3.3: Global error