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

TL;DR

This work tackles the challenge of simulating multi-dimensional $M$-coupled nonlinear Schrödinger equations, where analytical solutions are scarce, by deploying two high-order time-stepping schemes—fourth-order exponential time-differencing RK and fourth-order integrating factor RK—alongside a Fourier spectral discretization of space. Both methods achieve fourth-order temporal accuracy and spectral spatial convergence, with the Krogstad-P22 variant consistently delivering superior accuracy and efficiency across 1D–3D test problems under periodic, Dirichlet, and Neumann boundaries. The simulations demonstrate conservation of mass and energy to high precision and reveal the methods’ capability to capture complex soliton interactions, including elastic and inelastic collisions, across multi-dimensional settings. The results suggest that the Krogstad-P22 approach is particularly well-suited for long-time, high-dimensional nonlinear wave dynamics in applications such as nonlinear optics and Bose-Einstein condensates, providing a robust tool for exploring multi-component nonlinear phenomena.

Abstract

Coupled nonlinear Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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 Schrödinger 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.

Figures (23)

• Figure 1: Stability regions for equispaced (a) $y\in \mathbb R^-$ and (b) $y\in [-2\pi i,2\pi i]$.
• Figure 2: Temporal convergence of the Krogstad-P22 method vs IFRK4-P13 under various boundary conditions
• Figure 3: Comparison of the computational efficiency of Krogstad-P22 and IFRK4-P13 methods
• Figure 4: Spatial convergence of the Fourier spectral method in combination with Krogstad-P22 and IFRK4-P13 under various boundary conditions
• Figure 5: Mass conservation comparison under various boundary conditions.

Theorems & Definitions (2)

• Definition 1: Pad$\acute{\mathrm{e}}$-(r,s) Rational Approximation
• Definition 2: Big-$\mathcal{O}$ Time Complexity