Efficiently and accurately simulating multi-dimensional M-coupled nonlinear Schr\"odinger equations with fourth-order time integrators and Fourier spectral method

Abstract

Coupled nonlinear Schr\"odinger equations model various physical phenomena, such as wave propagation in nonlinear optics, multi-component Bose-Einstein condensates, and shallow water waves. Despite their extensive applications, analytical solutions of coupled nonlinear Schr\"odinger equations are widely either unknown or challenging to compute, prompting the need for stable and efficient numerical methods to understand the nonlinear phenomenon and complex dynamics of systems governed by coupled nonlinear Schr\"odinger equations. This paper explores the use of the fourth-order Runge-Kutta based exponential time-differencing and integrating factor methods combined with the Fourier spectral method to simulate multi-dimensional M-coupled nonlinear Schr\"odinger equations. The theoretical derivation and stability of the methods, as well as the runtime complexity of the algorithms used for their implementation, are examined. Numerical experiments are performed on systems of two and four multi-dimensional coupled nonlinear Schr\"odinger equations. 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, and both may be considered further in modeling more nonlinear dynamics in future work.