Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods

In this paper, various split step spectral (SSSP) schemes are proposed for N-coupled nonlinear Schrödinger (N  -CNLS) equations, especially for the systems with N⩾3N⩾3 for which numerical studies are few. These schemes are spectrally accurate in space, and sth (s = 1, 2, 4, 6, 8) order in time. They are proved to be conservative and to admit the exact plane wave solution. Extensive numerical experiments are carried out for the 3- and 4-CNLS systems to confirm the theoretical analysis. Half of the schemes are shown to possess wonderful ability of capturing high-frequency waves. However, the eighth-order schemes which seems to be optimal fail in this test. Accuracy and efficiency of the schemes are compared with each other, and the high-order schemes exhibit better. Since the eighth-order schemes are not better than the sixth-order ones, it is believed that constructing too much high-order schemes are unnecessary. Moreover, interactions of two-soliton solutions are also well simulated by the fourth-order and the sixth-order schemes, so the two methods are sufficient for use.