Effect of Fiber Self Phase Modulation on the Splitting Error using the Strang Formulas

The generalized nonlinear Schr dinger equation describes the different physical phenomena encountered when ultrashort pulses propagate through dispersive and nonlinear fibers. If the pulse duration is of picoseconds order, the nonlinear Schr dinger equation can be simplified. However the analytical solution remains inaccessible except for some special cases like soliton. The symmetric split-step Fourier method (S-SSFM) which is derived from the Strang formulas, subdivides the global propagation distance into small steps of length h to calculate the numerical solution of this equation. By using only the fact that the dispersive and nonlinear operators do not commute the Baker-Campbell-Hausdorff formula shows that the global relative error of this method is O(h2). Our numerical simulation results show that this error depends also on the self phase modulation nonlinear term. For this purpose, we employ in this work an explicit representation of the nonlinear operator and we present four implementations: the S-SSFM1, S-SSFM2, T-SM1 and T-SM2 obtained respectively from some weighting coefficients (c0, c1) = (0, 1), (c0, c1) = (1, 0), (c0,c1,c2) = (-1,1,1) and (c0,c1,c2) = (1,-1,1). Thus, we have computed for an input Gaussian pulse, the numerical solutions and the global relative errors for each implementation. As results, the estimated slopes of the linear variations of the global relative errors allow showing that the S-SSFM1 and T-SM1 errors are O(h), S-SSFM2 and T-SM2 errors are O(h2); furthermore, the S-SSFM is more accurate than the T-SM. In order to obtain an indicator of accuracies, we present the variations of the global relative errors for some values of the propagation length of the fiber.