Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations

The coupled nonlinear Schrödinger equations are highly used in modeling the various phenomena in nonlinear fiber optics, like propagation of pulses. Efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system. In this paper, we introduce a new version of the Cox and Matthews third order exponential time differencing Runge–Kutta (ETD3RK) scheme based on the (1, 2)-Padé approximation to the exponential function. In addition, we present its local extrapolation form to improve temporal accuracy to the fourth order. The developed scheme and its extrapolation are seen to be strongly stable, which have ability to damp spurious oscillations caused by high frequency components in the solution. A computationally efficient algorithm of the new scheme, based on a partial fraction splitting technique is presented. In order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear Schrödinger equations and performed several numerical experiments on them. The numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long-range solitary solutions given by coupled nonlinear Schrödinger equations and conserved the conserved quantities mass and energy exactly, to at least five decimal places.