Efficiently and accurately simulating multi-dimensional M-coupled nonlinear Schrödinger equations with fourth-order time integrators and Fourier pseudo-spectral method

Abstract

Coupled nonlinear Schrödinger equations (CNLSEs) model various physical phenomena, such as wave propagation in nonlinear optics. Despite their extensive applications, analytical solutions of CNLSEs are widely either unknown or challenging to compute, prompting the need for stable and efficient numerical methods to understand the nonlinear phenomenon governed by CNLSEs. This manuscript explores the use of the fourth-order Runge–Kutta based exponential time-differencing and integrating factor methods combined with the Fourier pseudo-spectral method to simulate multi-dimensional M-CNLSEs. The theoretical derivation and stability analysis of the methods, along with the runtime complexity of the algorithms used, are examined. Numerical experiments are performed on systems of two and four multi-dimensional CNLSEs. It is demonstrated by the results that both methods effectively conserve mass and energy while maintaining fourth-order temporal and spectral spatial convergence. Overall, it is shown by the numerical results that the exponential time-differencing method outperforms the integrating factor method in this application.