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

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.