Uniform error bound of an explicit exponential wave integrator Fourier pseudo-spectral method for the long-time dynamics of the nonlinear Schrödinger equation with wave operator

This paper presents a novel exponential wave integral Fourier pseudo-spectral method tailored for solving the nonlinear Schrödinger equation with a wave operator (NLSW) under conditions of weak nonlinearity, where 0<ϵ≤1. The method is an explicit, time-reversible and two-level scheme that accurately approximates both the function u and its time derivative ∂tu at each computational step. A central feature of this approach is the use of the trapezoidal rule, which facilitates the derivation of a uniform error estimate valid up to time T=O(ϵ−β) with 0≤β≤2. The error is quantified as ϵ2−βτ2+hm−1 in the discrete H1 norm, and this estimate applies equally to un (the discrete approximation of u) and u˙n (the discrete approximation of ∂tu), regardless of the discrete grid parameters. Theoretical findings are further substantiated by numerical experiments, which confirm the accuracy and reliability of the proposed method.