Uniform Error Estimates of an Energy‐Preserving Exponential Wave Integrator for the Nonlinear SchröDinger Equation With Wave Operator

ABSTRACT We propose a time‐symmetric and energy‐preserving exponential wave integrator Fourier pseudo‐spectral (EWIFP) method for the nonlinear Schrödinger equation with wave operator (NLSW). As , the NLSW tends towards the nonlinear Schrödinger equation (NLSE). The solution to the NLSW diverges from that of the NLSE by a function that oscillates in time with a wavelength of and amplitudes of for ill‐prepared initial conditions and for well‐prepared initial conditions, respectively. The proposed scheme demonstrates temporal symmetry and exact preservation of discrete energy conservation laws. We demonstrate that the proposed EWIFP method achieves uniform error bounds at . Here, is a parameter related to the regularity assumptions, while and represent the mesh size and time step, respectively. This implies that the EWIFP method attains uniform accuracy of order one with respect to for both initial data, which are ill‐prepared or well‐prepared, while also achieving spectral accuracy in space for both cases. Numerical experiments are provided to substantiate our theoretical error analysis.