Time-relaxation structure-preserving explicit low-regularity integrators for the nonlinear Schrödinger equation

2025/10/03 by Hang Li, Xicui Li, Li, Hang +5 · 2 citations
Mathematics · Physics and Astronomy · #35Q55 #65M12 #65M15 #65T50 #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2510.02963

openalex publication_date 2025/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose and rigorously analyze a novel family of explicit low-regularity exponential integrators for the nonlinear Schrödinger (NLS) equation, based on a time-relaxation framework. The methods combine a resonance-based scheme for the twisted variable with a dynamically adjusted relaxation parameter that guarantees exact mass conservation. Unlike existing symmetric or structure-preserving low-regularity integrators, which are typically implicit and computationally expensive, the proposed methods are fully explicit, mass-conserving, and well-suited for solutions with low regularity. Furthermore, the schemes can be naturally extended to a broad class of evolution equations exhibiting the structure of strongly continuous contraction semigroups. Numerical results demonstrate the accuracy, robustness, and excellent long-time behavior of the methods under low-regularity conditions.

Citations

• Explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation
• Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations
• A symmetric low-regularity integrator for the nonlinear Schrödinger equation
• Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes
• Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation
• A second order low-regularity integrator for the nonlinear Schrödinger equation
• A fully discrete low-regularity integrator for the nonlinear Schrödinger equation
• A fully discrete low-regularity integrator for the 1D periodic cubic nonlinear Schrödinger equation
• A general framework of low regularity integrators
• Relaxation Runge-Kutta Methods: Conservation and stability for\n Inner-Product Norms
• A Fourier integrator for the cubic nonlinear Schr "odinger equation\n with rough initial data
• Low regularity exponential-type integrators for semilinear Schrödinger equations

Cited by

• Conserving mass, momentum, and energy for the Benjamin-Bona-Mahony, Korteweg-de Vries, and nonlinear Schrödinger equations
• High-order mass- and energy-conserving methods for the nonlinear Schrödinger equation and its hyperbolization

Related

• Explicit symmetric low-regularity integrators for the nonlinear Schrödinger equation
• A symmetric low-regularity integrator for the nonlinear Schrödinger equation
• Accurate Solution of the Nonlinear Schrödinger Equation via Conservative Multiple-Relaxation ImEx Methods
• Low regularity exponential-type integrators for semilinear Schrödinger equations
• Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations