An accurate Fourier splitting scheme for solving the cubic quintic complex Ginzburg–Landau equation

• We proposed a split–splitting scheme for solving the cubic quintic complex Ginzburg–Landau equation (CQCGLE).
• The proposed scheme was derived using the same splitting technic used to solve the nonlinear Schrödinger equation discussed.
• We then seek other possibilities for solving CQCGLE by decorticating the problem into more sub-problems.
• The resulted schemes were compared with other methods in terms of accuracy and execution time.
• We then give different examples for solving the CQCGLE using our approach.

In this paper, we present a splitting scheme for the pseudo-spectral numerical method namely the Split-Step Fourier method (SSFM), in our approach we expand the exponential term in a manner that a succession of linear and nonlinear terms are distributed uniformly along one step size, the splitting will be performed symmetrically, this new scheme will be tested on one of the most used nonlinear partial deferential equation in optics, namely the cubic quintic complex Ginzburg–Landau (CQCGL) equation, in this work we demonstrate that the accuracy of the Split Step Fourier method scheme can be improved by expanding and distributing it in small parts within one step.