Efficiency of exponential time differencing schemes for nonlinear Schrödinger equations

The nonlinear Schrödinger (NLS) equation and its higher order extension (HONLS equation) are used extensively in modeling various phenomena in nonlinear optics and wave mechanics. Fast and accurate nonlinear numerical techniques are needed for further analysis of these models. In this paper, we compare the efficiency of existing Fourier split-step versus exponential time differencing methods in solving the NLS and HONLS equations. Soliton, Stokes wave, large amplitude multiple mode breather, and N-phase solution initial data are considered. To determine the computational efficiency we determine the minimum CPU time required for a given scheme to achieve a specified accuracy in the solution u(x, t) (when an analytical solution is available for comparison) or in one of the associated invariants of the system. Numerical simulations of both the NLS and HONLS equations show that for the initial data considered, the exponential time differencing scheme is computationally more efficient than the Fourier split-step method. Depending on the error measure used, the exponential scheme can be an order of magnitude more efficient than the split-step method.