Two completely explicit and unconditionally convergent Fourier pseudo-spectral methods for solving the nonlinear Schrödinger equation

Abstract

This paper aims to construct and analyze two new Fourier pseudo-spectral (FPS) methods for the general nonlinear Schrödinger (NLS) equation. The two FPS methods have two merits: unconditional convergence and complete explicitness in the practical computation. Further more, by introducing a modified mass functional and a modified energy functional, the two FPS methods are proved to preserve the total mass and energy in the discrete sense. Besides the standard energy method, the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio and initial value, we establish the optimal error estimate of the two FPS methods for solving the general NLS equation, while previous work just is valid for the cubic NLS equation and requires small initial value for the focusing case. These two FPS methods are proved to be spectrally accurate in space and second-order accurate in time, respectively. The analysis framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the NLS-type equations. We investigate the effect of the nonlinear term on the progression simulation of the plane wave, the conservation of the invariants and the effect of initial data on the blow-up solution via different parameters. Numerical results are reported to show the accuracy and efficiency of the proposed methods.