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

The generalized nonlinear Schrodinger 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 Schrodinger equation can be simp lified. However the analyt ical solution remains inaccessible except for some special cases like soliton. The symmetric split-step Fourier method (S-SSFM) which is derived fro m the St rang 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 formu la shows that the global relative error of this method is O(h 2 ). Our numerical simulation results show that this error depends also on the self phase modulation nonlinear term. For this purpose, we emp loy in this work an exp licit representation of the nonlinear operator and we present four imp lementations: the S-SSFM 1, S-SSFM 2, T-SM1 and T-SM 2 obtained respectively fro m