A High-Order Accurate and Time-Symmetric Exponential Wave Integrator for the Long-Term Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

Abstract

This paper is concerned with a novel exponential wave integrator Fourier pseudo-spectral method for simulating the long-term dynamics of the nonlinear Schrödinger equation with a wave operator under weak nonlinearity, characterized by \(0 < \varepsilon \ll 1\). The proposed method exhibits several notable features: it is symmetric or reversible in time, highly accurate in both temporal and spatial directions, and able to simultaneously approximate both \(u\) and its temporal derivative \(\partial_t u\) at each time step. A key technique in deriving our numerical method is to adopt the corrected trapezoidal rule, which enables the proposed scheme to achieve high-order temporal accuracy. An optimal and uniform error estimate is established for a time period up to \(T = O(\varepsilon^{-\beta})\) with \(\beta \in [0,2]\); the error bound is \((\varepsilon^{2-\beta}\tau^4 + h^{m-1})\) in the discrete \(H^1\) norm to the approximation of \(u\) and in the discrete \(L^2\) norm to the approximation of \(\partial_t u\) with time step \(\tau\) and mesh size \(h\). Numerical experiments are carried out to confirm our theoretical analysis.

The authors' abstract, as published at the source. Journal of Computational Mathematics, 2026 · DOI ↗